text:
# Speakers.

• Marcel Bökstedt, Aarhus University
• Gunnar Carlsson, Stanford University
• Ralph Cohen, Stanford University
• Herbert Edelsbrunner, Institute for Science and Technology, Austria
• Søren Galatius, Stanford University
• Kathryn Hess, EPFL
• Gerd Laures, Ruhr–Universität Bochum
• Arnold Levine, Institute for Advanced Study
• Ib Madsen, University of Copenhagen
• Edvard Moser, NTNU
• John Rognes, University of Oslo
• Dennis Sullivan, SUNY, Stony Brook
• Ulrike Tillmann, University of Oxford

# Organizing committee.

• Bjørn Ian Dundas, University of Bergen
• Bjørn Jahren, University of Oslo
• Erik Kjær Pedersen, University of Copenhagen
• Gereon Quick, NTNU
• Markus Szymik, NTNU
• Marius Thaule, NTNU

# Schedule.

All talks will take place in auditorium KJL2.

Wednesday November 30
Thursday December 1
Friday December 2
10:00 ‐ 10:10: Erik Kjær Pedersen: Welcome 09:30 ‐ 10:30: Ib Madsen 09:30 ‐ 10:30: Gunnar Carlsson
10:15 ‐ 11:15: Dennis Sullivan 10:45 ‐ 11:45: John Rognes 10:45 ‐ 11:45: Herbert Edelsbrunner
11:30 ‐ 12:30: Arnold Levine 11:45 ‐ 13:00: Lunch 11:45 ‐ 13:00: Lunch
12:30 ‐ 14:15: Lunch 13:00 ‐ 14:00: Søren Galatius 13:00 ‐ 14:00: Marcel Bökstedt
14:15 ‐ 15:15: Edvard Moser 14:15 ‐ 15:15: Gerd Laures 14:00 ‐ 14:30: Coffee
15:15 ‐ 16:00: Coffee 15:15 ‐ 16:00: Coffee 14:30 ‐ 15:30: Ulrike Tillmann
16:00 ‐ 17:00: Kathryn Hess 16:00 ‐ 17:00: Ralph Cohen
19:00: Conference dinner

# Abstracts.

All abstracts in one PDF file.

### Marcel Bökstedt: Spaces of vector sequences

We consider sequences of vectors $v_1,\dots,v_n$ in ${\mathrm C}^m$. These vectors are subject to the restrictions that for each $i$ the subsequence $v_i,v_{i + 1},\dots v_{i + m - 1}$ forms a basis for ${\mathrm C}^m$. We also fix the first $m$ and the last $m$ vectors in the sequence.

I will show that this approaches the space $\Omega(U_m/T^m)$ as $n$ goes to infinity, and use the limit case to study the cohomology of the unstable approximations. I will also try to argue that this is an interesting family of spaces. This is a mixture of work by me and by Simon Stolze, and much of it will appear in his PhD thesis.

### Gunnar Carlsson: Mayer–Vietoris and Products for Persistent Homology of Finite Metric Spaces

In this talk we will discuss methods for understanding the persistent homology of finite metric spaces which are decomposed into unions and products.

### Ralph Cohen: Comparing topological field theories: The string topology of a manifold and the Floer theory of its cotangent bundle

I will describe recent work of Costello, Lurie, and Kontsevich and his collaborators, which describe how 2D topological field theories are classified. In particular we describe two notions of duality among $A_\infty$-algebras and categories, and show how they give rise to field theories. These are referred to as “Calabi—Yau” structures. I’ll then describe joint work with S. Ganatra that uses these results to establish an equivalence between two chain complex valued topological field theories: the String Topology of a manifold and the Symplectic Field theory of its cotangent bundle. This generalizes results of Viterbo, Abbondandolo and Schwarz, Abouzaid, and others.

### Herbert Edelsbrunner: Distances and divergences in topological data analysis

Given a finite set in a metric space, the topological analysis assesses its multi-scale connectivity quantified in terms of a $1$-parameter family of homology groups. Going beyond metrics, we show that the basic tools of topological data analysis also apply when we measure dissimilarity with Bregman divergences. A particularly interesting case is the relative entropy whose infinitesimal version is known as the Fisher information metric. It relates to the Euclidean metric on the sphere and, perhaps surprisingly, the discrete Morse properties of random data behaves the same as in Euclidean space.

### Søren Galatius: Simplicial deformation rings and the homology of arithmetic groups

A classical result of Mazur asserts that any representation $\rho$ of a group (or pro-group) over a field admits a universal deformation. This is a representation over a complete local ring, whose reduction modulo the maximal ideal is identified with $\rho$, and which is in a suitable sense universal with these properties. I will discuss joint work with Akshay Venkatesh, in which we enrich this picture to simplicial rings. The classical deformation ring is recovered by taking $\pi_0$, but some higher homotopy groups may be non-trivial. Subject to some (partially established) conjectures, we prove that homology of certain arithmetic groups admit the structure of a free module structure over the homotopy groups of the simplicial deformation ring for certain Galois representations.

### Kathryn Hess: Topology meets neuroscience

I will present an overview of applications of topology to neuroscience on a wide range of scales, from the level of neurons to the level of brain regions. In particular I will describe in detail collaborations in progress with the Blue Brain Project on topological analysis of the structure and function of digitally reconstructed microconnectomes and on topological classification of neuron morphological types. I will then briefly sketch applications of topology to the analysis of brain imaging data.

### Gerd Laures: Trolls and other characteristic classes in TMF

Cannibalistic classes were first studied by Bott for real K-theory in the context of spherical fibrations. I will introduce the analogous classes for string bundles with values in various forms of TMF. I will provide some tools to compute them and will talk about possible applications.

### Arnold Levine: Cancer: The path from Evolutionary Changes to the Three Dimensional Structures of Proteins and Their Functions

The origins of cancers in humans derive from both inherited and somatic mutations in genes that are selected for growth and survival of cells under conditions where normal cells are controlled. Sequencing of the genetic information (DNA) from normal cells and cancerous cells, permits a catalog of mutations in those genes that contribute to cancerous growth. Genes encode the information for producing two-dimensional structures of proteins. Inherent in the sequence of amino acids in a protein is the information to fold or to be folded into a three–dimensional structure that provides a specific function for that protein. Mutations alter the three dimensional structure of proteins. Algorithms have been developed to predict the structural changes brought about by mutations, and the functional consequences of these changes in structure. Based upon this, drugs have been developed to restore the proper structure of a mutant, or altered, protein so as to restore function and destroy the cancer. These points will be illustrated by examining a large cohort of individuals who inherit a mutated cancer-causing version of the p53 gene. The natural history of individuals in this cohort over a lifetime will be explored and elucidated.

### Ib Madsen: Real algebraic K-theory

Real algebraic K-theory associates to an exact category with duality, e.g. the category of of projective modules with the usual duality, a $\mathbb{Z}/2$-equivariant spectrum (a real spectrum) and hence groups $K_{p,q}$ for each pair of integers. If the category is the category of projective modules over the ring of continuous functions on a compact space one obtains Atiyah's real K-groups. The talk will outline the construction and some of the theorems associated with it, including a Real version of the cyclotomic trace. At present, little is known in terms of calculations, so there is a lot to be done in making the Real cyclotomic trace an effective calculational tool. One would also like to generalize the whole theory to Waldhausen's A-theory. The lecture represents joint work with Lars Hesselholt.

### Edvard Moser: Grid cells and our sense of space

The entorhinal cortex and the hippocampus are elements of the brain's circuit for spatial navigation and memory. Interest in the functions of these brain areas was raised half a century ago, when a brain surgery affecting these areas left patient H.M. with a severe loss of episodic memory as well as an inability to navigate in space. This incidence motivated attempts to study the activity of neurons in the hippocampus of experimental animals and led, 15 years later, to the discovery of place cells — cells that fire if and only if animals are at certain locations. Over the past 15 years, we have explored the wider circuit of the mammalian positioning system. I will show that the entorhinal cortex contains grid cells — cells with firing fields that tile environments in a periodic hexagonal pattern, like an internal coordinate system — as well as cells that monitor direction, speed and local borders. Collectively these cells form the elements of a positioning system that dynamically monitors our changing location in the environment, and that may provide the spatial component of all episodic memories. Deficiencies in the function of this map may be at the core of neurological diseases where spatial orientation is affected, such as Alzheimer's disease.

### John Rognes: The Davis–Mahowald spectral sequence

$\DeclareMathOperator{sq}{Sq}$I will report on joint work with Robert R. Bruner. The cohomology of the subalgebra $A(2)$ of the mod $2$ Steenrod algebra, generated by $\sq^1$, $\sq^2$ and $\sq^4$, was calculated by May in his unpublished Ph.D. thesis, and in published form by Shimada and Iwai (1967). Davis and Mahowald (1982) gave a different approach to this calculation, using the more familiar cohomology groups of the subalgebra $A(1)$, generated by $\sq^1$ and $\sq^2$, as an intermediate step. We revisit the work of Davis and Mahowald to clarify the multiplicative structure present in their approach. As a result, we obtain a manageable understanding of the $E_2$-term of the Adams spectral sequence converging to the $2$-completed homotopy of the topological modular forms spectrum. A similar approach has proved to be useful in the analysis of the topological Hochschild homology of topological modular forms, as a step on the way to the topological periodic homology of that ring spectum.

### Dennis Sullivan: Manifolds with singularities

In 1953 Rene Thom showed that not all integral homology classes could be represented as images of top classes of closed manifolds. Classes up to six could be so represented by manifolds. But in dimension $7,8,9,\ldots$ manifolds with singularities would be required. It was fascinating to see what the singularities had to look like in the representatives of general homology classes.

An indirect argument 1970 lead to the picture that any homology class in dimension $7,8,9$ and $10$ could be represented by a map of a closed manifold with a codimension five sub manifold $S$ of non manifold points. The normal neighborhood of $S$ was a product of $S$ and the cone over $\mathbb{C}\mathrm{P}^2$, the complex projective plane.

Furthermore the homology class of $S$ represented a non zero mod three homology class inside the carrier of the cycle corresponding to a specific homology operation applied to the class being represented – the dual of the cohomology operation: Bockstein of Steenrods first $p$th power for the prime three.

Each consecutive set of four dimensions required adding a new singularity to those already needed. The picture was rich, pictorially understandable, and only sketched in the literature. The iterated singularity construction with many variants was developed carefully and correctly in nontrivial early work of Nils Baas. Once this machinery was in place one could rigorously construct many generalized homology theories with explicit geometric cycle representatives. To get a picture of Thom's remarkable discovery one only had to build an example with the correct homology groups for a point.

### Ulrike Tillmann: From moduli spaces of manifolds to K-theory

For mapping class groups of surfaces it is well-understood that their homology stability is closely related to the fact that they give rise to an infinite loop space. Indeed, they define an operad whose algebras group complete to infinite loop spaces.

In recent work with Basterra, Bobkova, Ponto and Yaekel we define operads with homology stability (OHS) more generally and prove that they are infinite loop space operads in the above sense. The strong homology stability results of Galatius and Randal-Williams for moduli spaces of manifolds can be used to construct examples of OHSs. As a consequence the map to K-theory defined by the action of the diffeomorphisms on the middle dimensional homology can be shown to be a map of infinite loop spaces.

# Registration.

Sending...
Thank you for registrering

Please fill out the form with your

• Name

• Institution

• Arrival

• Departure

• Special dietary requirements (if any) if you plan on attending the conference dinner.

Please note that we do not have funding available for travel and accommodation, except possibly for master and PhD students based in Norway.

Most participants, including speakers, will be staying at Scandic Nidelven. In addition to Scandic Nidelven we recommend Comfort Hotel Park and Comfort Hotel Trondheim.

Deadline for registration: October 20, 2016.

# Participants.

Name
Institution
Kristian Alfsvåg University of Bergen
Nils A. Baas NTNU
Alexander Berglund Stockholm University
Nello Blaser University of Bergen
Magnus Bakke Botnan TU Munich
Håvard Bakke Bjerkevik NTNU
Morten Brun University of Bergen
Marcel Bökstedt Aarhus University
Kaj Börjeson Stockholm University
Gunnar Carlsson Stanford University
Ralph Cohen Stanford University
Bjørn Ian Dundas University of Bergen
Herbert Edelsbrunner Institute of Science and Technology, Austria
Vegard Fjellbo University of Oslo
Kristoffer Føllesdal University of Bergen
Søren Galatius Stanford University
Alice Hedenlund University of Oslo
Kathryn Hess EPFL
Asaf Horev The Hebrew University of Jerusalem
Anders Husebø University of Bergen
Bjørn Jahren University of Oslo
Finn Faye Knudsen NTNU
Tore August Kro Østfold University College
Jonas Irgens Kylling University of Oslo
Gerd Laures Ruhr–Universität Bochum
Arnold Levine Institute for Advanced Study
Franz Luef NTNU
Tommy Lundemo University of Bergen
Edvard Moser NTNU
Espen Auseth Nielsen University of Copenhagen
Justin Noel University of Regensburg
Erik Kjær Pedersen University of Copenhagen
Stefano Piceghello University of Bergen
John Rognes University of Oslo
Erik Rybakken NTNU
Truls Bakkejord Ræder NTNU
Gereon Quick NTNU
Christian Schlichtkrull University of Bergen
Alexander Schmeding NTNU
Hans J. Skaug University of Bergen
Larry Smith University of Göttingen
Gard Spreemann EPFL
Eldar Straume NTNU
Johan Steen NTNU
Dennis Sullivan SUNY, Stony Brook
Markus Szymik NTNU
Marius Thaule NTNU
Sebastian Thyssen Ruhr–Universität Bochum
Ulrike Tillmann University of Oxford
Tam Thanh Truong University of Bergen
Glen Wilson University of Oslo
Christian Wimmer University of Bonn
Steffen Wittkamp Neusta data science lab
Lior Yanovski The Hebrew University of Jerusalem
Stephanie Ziegenhagen KTH


text:
# Linear Combination of Normal Distributions

I'm just playing around with RandomVariate. Right now I can't really figure out why a linear combination with weights 0.5 of two identical normal distributions is not yielding the same distribution. 0.5*N($\mu$,$\sigma$)+0.5*N($\mu$,$\sigma$) should yield N($\mu$,$\sigma$).

This fails if I use a Monte Carlo approach.

    Map[(
\[Mu] = 1;
\[Sigma] = 0.4;
p1 = RandomVariate[NormalDistribution[\[Mu], \[Sigma]], 10^4];
p2 = RandomVariate[NormalDistribution[\[Mu], \[Sigma]], 10^4];
FindDistribution[0.5*p1 + 0.5*p2]
) &, Range[10]]

I'm pretty sure Mathematica is doing everything right. Still I can't really figure out why the StandardDeviation is far away from 0.4.

• Check TransformedDistribution[ 1/2 x + 1/2 y, {x \[Distributed] NormalDistribution[\[Mu], \[Sigma]], y \[Distributed] NormalDistribution[\[Mu], \[Sigma]]}] and see if you agree with its output. – b.gates.you.know.what Aug 21 '18 at 8:38
• @b.gatesucks Thanks for your comment. TransformedDistribution yields practically the same result as my approach. – RMMA Aug 21 '18 at 8:56
• It would help if you were more specific with your terms. Are you interested in the linear combination of two normally distributed random variables, say, $Z=X/2+Y/2$? Or do are you interested in a mixture distribution where there are 3 random variables: $Z=\alpha X+(1-\alpha)Y$ where $X$ and $Y$ are normally distributed random variables (possibly independent of each other) and another Bernoulli random variable $\alpha$ that has $Pr(\alpha=1)=p$ and $Pr(\alpha=0)=1-p$ ? – JimB Aug 21 '18 at 21:07

Formular for the variance of the linear combination of two randomvariables:

$\operatorname{Var}( a \,X + b \, Y) = a^2 \, \operatorname{Var}(X) + b^2 \, \operatorname{Var}(Y) + 2 \, a \, b \operatorname{Cov}(X,Y)$

For the standard deviations, you have to take the square root.

Your random variables are independent and have variances 0.4^2 and a = b = 0.5. So the resulting standard deviation is

Sqrt[a^2 0.4^2 + b^2 0.4^2 + 2 a b 0.]

0.282843

That's quite exactly what you obtain empirically from FindDistribution.

Lesson to learn: The distribution of the sum is not the sum of the distributions.

• Thanks, we are right. – RMMA Aug 21 '18 at 9:19

I think the problem is the way you find the new sampled distribution. This is what you should do for testing you expression 0.5*N(μ,σ)+0.5*N(μ,σ) ~ N(μ,σ):

Map[(\[Mu] = 1;
\[Sigma] = 0.4;
p1 = RandomVariate[NormalDistribution[\[Mu], \[Sigma]], 10^4];
p2 = RandomVariate[NormalDistribution[\[Mu], \[Sigma]], 10^4];
{Mean@#, StandardDeviation@#} &@
MixtureDistribution[{0.5, 0.5}, {FindDistribution[p1],
FindDistribution[p2]}]
) &, Range[10]]

{{1.00193, 0.39969}, {1.00019, 0.402075}, {1.00143, 0.402793}, {0.997161, 0.399242}, {1.00011, 0.400222}, {1.00281, 0.401593}, {1.00001, 0.403046}, {1.00436, 0.400939}, {0.998176, 0.402365}, {1.00206, 0.396431}}

As you can see mean values and std devs are consistent with what you would expect.

• I do not really aggree, because what I want to get is a linear combination of normal distribution. This linear combination should be normally distributed itself. The mixture distribution is generally not a normal distribution. – RMMA Aug 21 '18 at 8:55
• @RMMA from MMA documentation, it seems like MixtureDistribution[{w1,w2},{dist1,dist2}] does exactly what you are asking for: 0.5*N(μ,σ)+0.5*N(μ,σ). In your code you are not summing the distributions divided by 2, while you are summing the "0.5*data" sampled from the distributions and then you compute the new distribution from this result, which is not the same thing – Fraccalo Aug 21 '18 at 8:59
• It does and I appreciate your answer. Ultimately I would like to calculate the linear combination of arbritray normal distributions and I think MixtureDistribution is not the right way to go in this case. A quick look at the PDF shows that the MixtureDistribution of two normal distributions is generally no normal distribution. Contrary the linear combination of two normal distribution should be a normal distribution. – RMMA Aug 21 '18 at 9:06
• Oh yes, I understand what you mean now, you want the general case (where mu and sigma might be different for the different distributions). In such a case I understand that the mixture doesn't help you much – Fraccalo Aug 21 '18 at 9:11

format: plain


text:
## Lebesgue measurable subsets of [0,1]

If $\displaystyle A$ is a Lebesgue measurable subset of [0,1] such that $\displaystyle \lambda (A)>0$. I need to show that there are Lebesgue measurable subsets $\displaystyle A_{s},0\leq s \leq \lambda(A)$ so that

(a) $\displaystyle A_{r}\subseteq A_{s}$ if $\displaystyle r\leq s$,
(b) $\displaystyle \lambda (A_{s})=s$ for any $\displaystyle 0\leq s \leq \lambda(A)$.

I started with $\displaystyle A_{s}=A\cap [0,s]$ and I can verify (a) easily. But I only have (b) if $\displaystyle [0,s]\subseteq A$ since
$\displaystyle \lambda (A_{s})=\lambda ([0,s])=s$

But (b) is not true if [0,s] is not a subset of $\displaystyle A$.

How can I modify my construction of the set $\displaystyle A_{s}$?

The s in this problem is a bit confusing for me as it is a measure and subscript at the same time. I don't really understand the requirement in (b). Any idea?


text:
Search
Question: Genes listed in GO categories
0
8 days ago by
frezende0
frezende0 wrote:

Hi everyone,

I’m testing for GO categories enrichment using goseq (version 1.32.0) under two different approaches (as described below) and I’m getting different numbers of genes per GO category.

### Approach 1 ###

> library(goseq)
> gene.vector = as.integer(assayed.genes%in%de.genes)  #19,792 total genes (502 DE genes)
> pwf = nullp(gene.vector, "bosTau4", "ensGene")
> GO.hiper = goseq(pwf, "bosTau4", "ensGene")
> GO.hiper[GO.hiper$category=="GO:0005509",] category over_represented_pvalue under_represented_pvalue numDEInCat numInCat term ontology 1788 GO:0005509 0.01579858 0.9942288 10 182 calcium ion binding MF Even if I force the inclusion of genes without categories in the test, the number of genes per category is the same! > GO.hiper = goseq(pwf, "bosTau4", "ensGene", use_genes_without_cat = TRUE) > GO.hiper[GO.hiper$category=="GO:0005509",]
category over_represented_pvalue under_represented_pvalue numDEInCat numInCat  term ontology
1788 GO:0005509       0.02582693                0.9895319         10      182 calcium ion binding       MF

### Approach 2 ###

> library(biomaRt)
> genome = useMart(biomart = "ENSEMBL_MART_ENSEMBL", dataset = "btaurus_gene_ensembl", host = "www.ensembl.org")
> Ca.gene.bio <- getBM(attributes=c("ensembl_gene_id", "start_position", "end_position", "chromosome_name", "external_gene_name"), filters = 'go', values = 'GO:0005509', mart = genome)  #589 genes
> geneTOcat = data.frame(cbind(Ca.gene.bio$ensembl_gene_id,"GO:0005509")) > GO.hiper = goseq(pwf, "bosTau4", "ensGene", gene2cat = geneTOcat, use_genes_without_cat = TRUE) > GO.hiper[GO.hiper$category=="GO:0005509",]
category over_represented_pvalue under_represented_pvalue numDEInCat numInCat  term ontology
1 GO:0005509              0.02445965                0.9861623         23      520 calcium ion binding       MF

Does the goseq use GO.db package (GO.db_3.6.0) to retrieve the genes listed in GO categories?

Is the GO annotation version used by goseq the latest one available on BioMart (biomaRt_2.36.1)?

It seems that goseq is not retrieving all the genes listed in a GO category. How can I fix it in the first approach?

Thank you for taking the time to read my question.

Have a nice weekend!

Fernanda Rezende

University of Florida, USA

modified 1 day ago by Gordon Smyth33k • written 8 days ago by frezende0
1
6 days ago by
United States
James W. MacDonald46k wrote:

You are, in situation #1, asking for all the Entrez Gene IDs that are associated with a given GO term. And in situation #2 you are asking for all the Ensembl Gene IDs that are associated with a given GO term. These are not the same thing! If you ask two different groups the same question you shouldn't be too surprised if they give different answers.

Put a different way, you seem to think that there is a known set of genes and everybody thinks the set is the same. This is not true. There are any number of differences between the annotations you can get from NCBI as compared to what you can get from EBI/EMBL.

I was curious though about how 1) the number of Ensembl IDs is almost exactly a factor of 3 greater than the number of Entrez IDs, and 2) the over-representation p-values come out almost exactly the same. I suppose the factor of 3 is just a coincidence but I'm surprised the gene set size is so different, and given that difference, I'm surprised the p-values were so consistent between the two approaches.

I totally agree with your point about different sources providing quite distinct annotations, but are those two scenarios really comparing different gene IDs?  I don't know much about the goseq functions, but both examples provide the argument 'ensGene' which implies using the Ensembl IDs in both instances to me.

I more wonder if this may be more related to using an out-of-date genome/annotation.  The bosTau4 genome was released in 2007, and I think we're up to bosTau8 now.  I wonder if goseq/GO.db struggles when it tries to identify the GO terms, where as when using biomaRt you basically bypass that.

3

The argument isn't 'ensGene', that's the input for the 'genes' argument, which tells goseq what sort of ID you are using. If the OP were using Entrez Gene IDs, then the argument would be 'knownGene'.

The genome has very little to do with this, other than for getting the gene lengths. In my mind, annotations and genome builds are almost completely orthogonal, with genome builds having releases and being tied to a certain time point (and also having only to do with where things are in the genome), whereas annotations have almost nothing to do with genome builds, but instead have to do with what genes we think might be in the genome, and what those genes do, etc. The gene annotations are updated almost constantly, with things like RefSeq having weekly updates, whereas the genome builds are released and then remain almost static.

That's a long way of saying that the data in GO.db and org.Bt.eg.db are current as of early 2018, with the notable exception of the Ensembl data which appears to come from 2017, for whatever reason. And that's the data we are relying upon to match GO and Ensembl IDs.

The weird thing (to me) with goseq is that you can (and have to, in this case) use Ensembl Gene IDs, but if you don't pass in a gene2cat data.frame, then under the hood it will call getgo, which just dumps out the org.Bt.egGOALLEGS table from org.Bt.eg.db, which is a mapping from EG to GO, and then uses the org.Bt.egENSEMBL table to map EG to Ensembl IDs. Which is where the problem lies.

> z <- getBM("ensembl_gene_id", "go","GO:0005509", mart)
> gene.vector <- sample(0:1, length(z[,1]), TRUE)
> names(gene.vector) <- z[,1]
> pwf <- nullp(gene.vector, "bosTau4","ensGene")
DEgenes bias.data       pwf
ENSBTAG00000017746       0    4551.0 0.5106931
ENSBTAG00000002865       1     665.5 0.4503569
ENSBTAG00000007244       1    5812.0 0.5302726
ENSBTAG00000009696       1    2656.0 0.4812669
ENSBTAG00000011862       1    2858.0 0.4844038
ENSBTAG00000020657       1   13793.0 0.6541682
> dim(pwf)
[1] 589   3
> gene2cat = getgo(rownames(pwf), "bosTau4","ensGene")

> sum(sapply(gene2cat, is.null))
[1] 402
> length(gene2cat)
[1] 589

So there are two issues here. First, there are only 211 Entrez Gene IDs that map to GO:0005509, according to (evidently) NCBI and Geneontology, as of April this year. There are more that map from Ensembl IDs, but the way getgo works, using old code like toTable rather than select or mapIds, we lose out on bunches of EG -> Ensembl mappings:

> head(names(gene2cat))
[1] NA                   NA                   NA
[4] "ENSBTAG00000009696" NA                   NA

> huh2 <- mapIds(org.Bt.eg.db, row.names(pwf), "ENTREZID", "ENSEMBL")
'select()' returned 1:many mapping between keys and columns
> length(huh2)
[1] 589
ENSBTAG00000017746 ENSBTAG00000002865 ENSBTAG00000007244 ENSBTAG00000009696
"615883"           "541458"                 NA           "536607"
ENSBTAG00000011862 ENSBTAG00000020657
"616023"           "508251"

> sum(is.na(huh2))
[1] 64

1

James, I am not a goseq author and I haven't examined the code, but it seems to me that goseq is most likely doing the right thing.

You compute a vector huh2 of over 500 gene ids, but only 184 of those ids can be mapped to GO terms using the organism package. The entries that are NA in gene2cat but not NA in huh2 are exactly those ids that can't be mapped.

> MyGO <- org.Bt.egGO2ALLEGS
> Rkeys(MyGO) <- "GO:0005509"
> MyEG <- mappedLkeys(MyGO)
> length(MyEG)
[1] 205
> sum(huh2 %in% MyEG, na.rm=TRUE)
[1] 184

I think that's correct. Thanks for pointing that out. The only real issue here (IMO) then is that Ensembl IDs will be silently converted to Gene IDs to do the GO mapping if a gene2cat data.frame isn't supplied. IMO this is a fraught process! The OP has already documented that EBI/EMBL have quite different ideas about what GO terms should be attributed to a given Ensembl ID as compared to NCBI.

While there are some warnings about sticking with the same annotation service (particularly in the details section for getgo), both the example for getgo and the vignette seem to indicate that this is a non-issue, as Ensembl IDs are used for both the example for getgo, as well as the examples in the vignette. This seems to indicate that going across annotation services is a reasonable thing to do, whereas I would argue just the opposite.

I agree that sticking to native gene ids is desirable if one can do it, and I agree that the goseq documentation gives the impression of being Ensembl orientated whereas it actually uses the Bioc organism packages by default.

But do you really mean to say that mapping gene Ids is never a sensible thing to do? The Bioconductor organism packages are Entrez Gene based, so Bioconductor users and developers have no choice but to convert back to Entrez Gene Ids if they wish to use Bioconductor annotation infrastructure.

I think that GO annotation pretty much always involves mapping across annotation systems anyway. For mouse, the master GO annotation source is MGI, for drosophila it is flybase, so the NCBI and Ensembl annotation for those species are both translated from the original. A little bit of searching suggests that the bos Taurus GO associations may be UniProt based. For bos Taurus, I would guess that most of the GO annotation is based on mapping orthologs from other species anyway.

No, I am not saying that mapping across services is never a sensible thing to do. What I am saying is that it is a non-trivial thing to do, because it may have unexpected consequences. For example, in the original post to this thread the OP states that the Ensembl mappings for that GO term were for 520 genes. But after processing through the Gene IDs, we are down to 182.

This didn't have an effect on the results for this particular GO term, because there was a proportional loss of the DE genes as well as the nonDE genes for that category. Is that consistent? I sort of doubt it. To check we could simulate some data:

library(RMySQL)
library(goseq)
library(biomaRt)
mart <- useMart("ensembl","btaurus_gene_ensembl")
dbconn <- dbConnect(MySQL(), user="genome", host = "genome-mysql.soe.ucsc.edu", dbname = "bosTau4")
## get all Ensembl IDs for bosTau4
btens <- dbGetQuery(dbconn, "select distinct name2 from ensGene;")
btens2 <- rep(0, nrow(btens))
names(btens2) <- btens[,1]
set.seed(0xabeef)
## simulate 500 significant genes
sig.genes <- sample(1:nrow(btens), 500)
btens2[sig.genes] <- 1
## Ensembl mapping to GO
gene2cat <- getBM(c("ensembl_gene_id","go_id"), "ensembl_gene_id", as.character(btens[,1]), mart)
pwf <- nullp(btens2, "bosTau4", "ensGene")
z <- goseq(pwf, "bosTau4", "ensGene")
zz <- goseq(pwf, "bosTau4", "ensGene", gene2cat = gene2cat)
z2 <- goseq(pwf, "bosTau4", "ensGene", use_genes_without_cat = TRUE)
zz2 <- goseq(pwf, "bosTau4", "ensGene", gene2cat = gene2cat, use_genes_without_cat = TRUE)

A Venn diagram of the significantly over-represented GO terms (using unadjusted p < 0.05), where use_genes_without_cat = TRUE:

and if we only use those genes with a GO term:

That's IMO a big difference in results, based on a purely technical issue of how the genes were mapped to GO IDs.

Thanks for the comprehensive answer.  I didn't appreciate that under the hood the core relationship is always GOTerm-to-EntrezID and that the will always be another mapping required for Gene IDs from any other annotation.

1
1 day ago by
Gordon Smyth33k
Walter and Eliza Hall Institute of Medical Research, Melbourne, Australia
Gordon Smyth33k wrote:

Dear Fernanda,

Actually goseq is using up-to-date annotation.

Here is what's happening. If you supply your own gene2cat annotation, then goseq will use that. If you don't, then goseq uses the GO-gene associations provided in the relevant Bioconductor organism package, in this case org.Bt.eg.db. The Bioconductor organism packages are based on NCBI annotation and Entrez Gene Ids.

For reasons that we don't completely understand, Ensembl associates over 500 Ensembl genes with your GO term but NCBI associates only 200 Entrez Genes with the same GO term. So the difference between the two approach is not a fault of goseq, but just the difference between Ensembl and NCBI.

As a general rule, NCBI is more conservative than Ensembl. Ensembl tends to include more of everything: genes, transcripts, associations and so on. NCBI tends to include only well annotated features and associations whereas Ensembl tends to include more features for which there is less (but still some) evidence.

In this case, you will notice that the enrichment of your GO term is weakened rather than strengthened when you add in the extra Ensembl gene associations, because 10/182 is stronger enrichment than 23/520. In other words, the genes given by Ensembl but not NCBI were less likely to be in your DE list than genes given by both. Now I know nothing about bos Taurus or about this particular GO term, so this particular example could be coincidence. But, as a general principle, one needs to balance the greater comprehensiveness of Ensembl with the greater reliability of NCBI, and we often do see greater enrichment rates for pathway analyses using better annotated genes.

In this case, the difference between NCBI and Ensembl is surprisingly large. You have decide yourself when to use Ensembl (European) or NCBI (American) annotation. goseq allows you to do either. Since you decided to use Ensembl genes in the first place, it would be consistent to use Ensembl gene-GO associations. However the NCBI-based analysis provided by goseq in Approach1 is probably also good, for the reasons I have explained.


text:
# conf¶

## Loading a configuration¶

Various aspects of PyPhi’s behavior can be configured.

When PyPhi is imported, it checks for a YAML file named pyphi_config.yml in the current directory and automatically loads it if it exists; otherwise the default configuration is used.

The various settings are listed here with their defaults.

>>> import pyphi
>>> defaults = pyphi.config.defaults()

Print the config object to see the current settings:

>>> print(pyphi.config)
{ 'ASSUME_CUTS_CANNOT_CREATE_NEW_CONCEPTS': False,
'CACHE_SIAS': False,
'CACHE_POTENTIAL_PURVIEWS': True,
'CACHING_BACKEND': 'fs',
...

Setting can be changed on the fly by assigning them a new value:

>>> pyphi.config.PROGRESS_BARS = False

It is also possible to manually load a configuration file:

>>> pyphi.config.load_config_file('pyphi_config.yml')

Or load a dictionary of configuration values:

>>> pyphi.config.load_config_dict({'PRECISION': 1})

## Approximations and theoretical options¶

These settings control the algorithms PyPhi uses.

## Parallelization and system resources¶

These settings control how much processing power and memory is available for PyPhi to use. The default values may not be appropriate for your use-case or machine, so please check these settings before running anything. Otherwise, there is a risk that simulations might crash (potentially after running for a long time!), resulting in data loss.

## Memoization and caching¶

PyPhi provides a number of ways to cache intermediate results.

## Logging¶

These settings control how PyPhi handles log messages. Logs can be written to standard output, a file, both, or none. If these simple default controls are not flexible enough for you, you can override the entire logging configuration. See the documentation on Python’s logger for more information.

## The config API¶

class pyphi.conf.Option(default, values=None, on_change=None, doc=None)

A descriptor implementing PyPhi configuration options.

Parameters: Keyword Arguments: default – The default value of this Option. values (list) – Allowed values for this option. A ValueError will be raised if values is not None and the option is set to be a value not in the list. on_change (function) – Optional callback that is called when the value of the option is changed. The Config instance is passed as the only argument to the callback. doc (str) – Optional docstring for the option.
class pyphi.conf.ConfigMeta(cls_name, bases, namespace)

Metaclass for Config.

Responsible for setting the name of each Option when a subclass of Config is created; because Option objects are defined on the class, not the instance, their name should only be set once.

Python 3.6 handles this exact need with the special descriptor method __set_name__ (see PEP 487). We should use that once we drop support for 3.4 & 3.5.

class pyphi.conf.Config

Base configuration object.

See PyphiConfig for usage.

classmethod options()

Return a dictionary the Option objects for this config

defaults()

Return the default values of this configuration.

load_config_dict(dct)

Load a dictionary of configuration values.

load_config_file(filename)

Load config from a YAML file.

snapshot()

Return a snapshot of the current values of this configuration.

override(**new_values)

Decorator and context manager to override configuration values.

The initial configuration values are reset after the decorated function returns or the context manager completes it block, even if the function or block raises an exception. This is intended to be used by tests which require specific configuration values.

Example

>>> from pyphi import config
>>> @config.override(PRECISION=20000)
... def test_something():
...     assert config.PRECISION == 20000
...
>>> test_something()
>>> with config.override(PRECISION=100):
...     assert config.PRECISION == 100
...

pyphi.conf.configure_logging(conf)

Reconfigure PyPhi logging based on the current configuration.

class pyphi.conf.PyphiConfig

pyphi.config is an instance of this class.

ASSUME_CUTS_CANNOT_CREATE_NEW_CONCEPTS

default=False

In certain cases, making a cut can actually cause a previously reducible concept to become a proper, irreducible concept. Assuming this can never happen can increase performance significantly, however the obtained results are not strictly accurate.

CUT_ONE_APPROXIMATION

default=False

When determining the MIP for $$\Phi$$, this restricts the set of system cuts that are considered to only those that cut the inputs or outputs of a single node. This restricted set of cuts scales linearly with the size of the system; the full set of all possible bipartitions scales exponentially. This approximation is more likely to give theoretically accurate results with modular, sparsely-connected, or homogeneous networks.

MEASURE

default='EMD'

The measure to use when computing distances between repertoires and concepts. Users can dynamically register new measures with the pyphi.distance.measures.register decorator; see distance for examples. A full list of currently installed measures is available by calling print(pyphi.distance.measures.all()). Note that some measures cannot be used for calculating $$\Phi$$ because they are asymmetric.

PARALLEL_CONCEPT_EVALUATION

default=False

Controls whether concepts are evaluated in parallel when computing cause-effect structures.

PARALLEL_CUT_EVALUATION

default=True

Controls whether system cuts are evaluated in parallel, which is faster but requires more memory. If cuts are evaluated sequentially, only two SystemIrreducibilityAnalysis instances need to be in memory at once.

PARALLEL_COMPLEX_EVALUATION

default=False

Controls whether systems are evaluated in parallel when computing complexes.

NUMBER_OF_CORES

default=-1

Controls the number of CPU cores used to evaluate unidirectional cuts. Negative numbers count backwards from the total number of available cores, with -1 meaning ‘use all available cores.’

MAXIMUM_CACHE_MEMORY_PERCENTAGE

default=50

PyPhi employs several in-memory caches to speed up computation. However, these can quickly use a lot of memory for large networks or large numbers of them; to avoid thrashing, this setting limits the percentage of a system’s RAM that the caches can collectively use.

CACHE_SIAS

default=False

PyPhi is equipped with a transparent caching system for SystemIrreducibilityAnalysis objects which stores them as they are computed to avoid having to recompute them later. This makes it easy to play around interactively with the program, or to accumulate results with minimal effort. For larger projects, however, it is recommended that you manage the results explicitly, rather than relying on the cache. For this reason it is disabled by default.

CACHE_REPERTOIRES

default=True

PyPhi caches cause and effect repertoires. This greatly improves speed, but can consume a significant amount of memory. If you are experiencing memory issues, try disabling this.

CACHE_POTENTIAL_PURVIEWS

default=True

Controls whether the potential purviews of mechanisms of a network are cached. Caching speeds up computations by not recomputing expensive reducibility checks, but uses additional memory.

CLEAR_SUBSYSTEM_CACHES_AFTER_COMPUTING_SIA

default=False

Controls whether a Subsystem’s repertoire and MICE caches are cleared with clear_caches() after computing the SystemIrreducibilityAnalysis. If you don’t need to do any more computations after running sia(), then enabling this may help conserve memory.

CACHING_BACKEND

default='fs'

Controls whether precomputed results are stored and read from a local filesystem-based cache in the current directory or from a database. Set this to 'fs' for the filesystem, 'db' for the database.

FS_CACHE_VERBOSITY

default=0

Controls how much caching information is printed if the filesystem cache is used. Takes a value between 0 and 11.

FS_CACHE_DIRECTORY

default='__pyphi_cache__'

If the filesystem is used for caching, the cache will be stored in this directory. This directory can be copied and moved around if you want to reuse results e.g. on a another computer, but it must be in the same directory from which Python is being run.

MONGODB_CONFIG

default={'collection_name' -- 'cache', 'host' -- 'localhost', 'port': 27017, 'database_name': 'pyphi'}

Set the configuration for the MongoDB database backend (only has an effect if CACHING_BACKEND is 'db').

REDIS_CACHE

default=False

Specifies whether to use Redis to cache MaximallyIrreducibleCauseOrEffect.

REDIS_CONFIG

default={'host' -- 'localhost', 'port' -- 6379}

Configure the Redis database backend. These are the defaults in the provided redis.conf file.

LOG_FILE

default='pyphi.log', on_change=configure_logging

Controls the name of the log file.

LOG_FILE_LEVEL

default='INFO', on_change=configure_logging

Controls the level of log messages written to the log file. This setting has the same possible values as LOG_STDOUT_LEVEL.

LOG_STDOUT_LEVEL

default='WARNING', on_change=configure_logging

Controls the level of log messages written to standard output. Can be one of 'DEBUG', 'INFO', 'WARNING', 'ERROR', 'CRITICAL', or None. 'DEBUG' is the least restrictive level and will show the most log messages. 'CRITICAL' is the most restrictive level and will only display information about fatal errors. If set to None, logging to standard output will be disabled entirely.

LOG_CONFIG_ON_IMPORT

default=True

Controls whether the configuration is printed when PyPhi is imported.

Tip

If this is enabled and LOG_FILE_LEVEL is INFO or higher, then the log file can serve as an automatic record of which configuration settings you used to obtain results.

PROGRESS_BARS

default=True

Controls whether to show progress bars on the console.

Tip

If you are iterating over many systems rather than doing one long-running calculation, consider disabling this for speed.

PRECISION

default=6

If MEASURE is EMD, then the Earth Mover’s Distance is calculated with an external C++ library that a numerical optimizer to find a good approximation. Consequently, systems with analytically zero $$\Phi$$ will sometimes be numerically found to have a small but non-zero amount. This setting controls the number of decimal places to which PyPhi will consider EMD calculations accurate. Values of $$\Phi$$ lower than 10e-PRECISION will be considered insignificant and treated as zero. The default value is about as accurate as the EMD computations get.

VALIDATE_SUBSYSTEM_STATES

default=True

Controls whether PyPhi checks if the subsystems’s state is possible (reachable with nonzero probability from some previous state), given the subsystem’s TPM (which is conditioned on background conditions). If this is turned off, then calculated $$\Phi$$ values may not be valid, since they may be associated with a subsystem that could never be in the given state.

VALIDATE_CONDITIONAL_INDEPENDENCE

default=True

Controls whether PyPhi checks if a system’s TPM is conditionally independent.

SINGLE_MICRO_NODES_WITH_SELFLOOPS_HAVE_PHI

default=False

If set to True, the $$\Phi$$ value of single micro-node subsystems is the difference between their unpartitioned CauseEffectStructure (a single concept) and the null concept. If set to False, their $$\Phi$$ is defined to be zero. Single macro-node subsystems may always be cut, regardless of circumstances.

REPR_VERBOSITY

default=2, values=[0, 1, 2]

Controls the verbosity of __repr__ methods on PyPhi objects. Can be set to 0, 1, or 2. If set to 1, calling repr on PyPhi objects will return pretty-formatted and legible strings, excluding repertoires. If set to 2, repr calls also include repertoires.

Although this breaks the convention that __repr__ methods should return a representation which can reconstruct the object, readable representations are convenient since the Python REPL calls repr to represent all objects in the shell and PyPhi is often used interactively with the REPL. If set to 0, repr returns more traditional object representations.

PRINT_FRACTIONS

default=True

Controls whether numbers in a repr are printed as fractions. Numbers are still printed as decimals if the fraction’s denominator would be large. This only has an effect if REPR_VERBOSITY > 0.

PARTITION_TYPE

default='BI'

Controls the type of partition used for $$\varphi$$ computations.

If set to 'BI', partitions will have two parts.

If set to 'TRI', partitions will have three parts. In addition, computations will only consider partitions that strictly partition the mechanism the mechanism. That is, for the mechanism (A, B) and purview (B, C, D) the partition:

A,B    ∅
─── ✕ ───
B    C,D

is not considered, but:

 A     B
─── ✕ ───
B    C,D

is. The following is also valid:

A,B     ∅
─── ✕ ─────
∅    B,C,D

In addition, this setting introduces “wedge” tripartitions of the form:

 A     B     ∅
─── ✕ ─── ✕ ───
B     C     D

where the mechanism in the third part is always empty.

In addition, in the case of a $$\varphi$$-tie when computing a MaximallyIrreducibleCause or MaximallyIrreducibleEffect, The 'TRIPARTITION' setting choses the MIP with smallest purview instead of the largest (which is the default).

Finally, if set to 'ALL', all possible partitions will be tested.

PICK_SMALLEST_PURVIEW

default=False

When computing a MaximallyIrreducibleCause or MaximallyIrreducibleEffect, it is possible for several MIPs to have the same $$\varphi$$ value. If this setting is set to True the MIP with the smallest purview is chosen; otherwise, the one with largest purview is chosen.

USE_SMALL_PHI_DIFFERENCE_FOR_CES_DISTANCE

default=False

If set to True, the distance between cause-effect structures (when computing a SystemIrreducibilityAnalysis) is calculated using the difference between the sum of $$\varphi$$ in the cause-effect structures instead of the extended EMD.

SYSTEM_CUTS

default='3.0_STYLE', values=['3.0_STYLE', 'CONCEPT_STYLE']

If set to '3.0_STYLE', then traditional IIT 3.0 cuts will be used when computing $$\Phi$$. If set to 'CONCEPT_STYLE', then experimental concept-style system cuts will be used instead.

log()

Log current settings.


text:
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Get Full Access to Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) - 3 Edition - Chapter 5 - Problem 5.5.46
Get Full Access to Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) - 3 Edition - Chapter 5 - Problem 5.5.46

×

# Classify each of the quadratic forms in Exercises 4148 as positive definite, positive

ISBN: 9780538735452 298

## Solution for problem 5.5.46 Chapter 5

Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) | 3rd Edition

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Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) | 3rd Edition

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Problem 5.5.46

Classify each of the quadratic forms in Exercises 4148 as positive definite, positive semidefinite, negative definite, negative semidefinite, or indefinite.

Step-by-Step Solution:
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Elementary Statistical Methods Notes: January 11-15, 2016 Chapter 1: Intro to Statistics Section 1.1 Statistics is the science of collecting, describing, and interpreting data. There are two types of Statistics:  Descriptive Statistics (Chapters 2-4)  Inferential Statistics (Chapters 8-11) (Chapters 5-7 cover probability)  Chapter 5: Probability  Chapter 6: Discrete probability distribution  Chapter 7: Continuous probability distribution Descriptive Statistics includes the collection, presentation, and description of sample data (histograms, bar graphs, pie charts, etc…) Inferential Statistics consists of procedures to make inferences or conclusions about population characteristics from the info contained in the sample. A variable is a characteristic that changes or varies over time and/or for different individuals or objects under consideration (e.g. test scores, sales, etc…) Example: Body temperature is a variable that changes over time within a single person, it also varies from person to person. Data are observations (e.g. measurements) that have been collected. A population is the set of all individuals or objects or measurements whose properties are to be analyzed. A sample is a small part of the total population. A parameter is a numerical measurement describing some characteristics of an entire population. Example: The average age at time of admission for all students attending ECU *Greek letters are used to symbolize parameters:  Mu = mean

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## Discover and learn what students are asking

Calculus: Early Transcendental Functions : Parametric Equations and Calculus
?In Exercises 1-4, find $$d y / d x$$. $$x=2 e^{\theta}, y=e^{-\theta / 2}$$

Statistics: Informed Decisions Using Data : Estimating a Population Standard Deviation
?True or False: A confidence interval for a population standard deviation is of the form point estimate ± margin of error.

Statistics: Informed Decisions Using Data : Inference about Two Population Standard Deviations
?In Problems 9–14, assume that the populations are normally distributed. Test the given hypothesis. 9. s1 ? s2 at the a = 0.05 level of signif

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$\text{Given a finite string }S\text{ of symbols }X\text{ and }O\text{, we write }\Delta(S)\text{for the number of } \\ X\text{'s in }S\text{ minus the number of }O\text{'s. For example, }\Delta(XOOXOOX) = -1\text{. We} \\ \text{call a string }S\textbf{ balanced}\text{ if every substring }T\text{ of (consecutive symbols of) }S \\ \text{has }-2 \leq \Delta(T) \leq 2\text{. Thus, }XOOXOOX\text{ is not balanced, since it contains the} \\ \text{substring }OOXOO\text{. Find, with proof, the number of balanced strings of length} \\ n\text{.}$
>>
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>>10468674
Define a partial string of S as a substring that contains the first k elements of S, where k is some natural number such that 1<k<n. A string S is balanced iff the maximum delta of the partial strings of S minus the minimum delta of the partial strings of S is between -2 and 2. The formula should be 2 * (1 + sum from i = 2 to n of 2^(ceil((n - i - 1)/2))). At the first part of the algorithm, you either pick X or O, and then you fill in with alternating symbols; you continue until i - 1 terms of the sequence have been filled in. i represents the part of the sequence where you duplicate the previous term, however, after duplication, you must pick the opposite symbol, and, from there, you may start filling in the rest of the sequence with either XOs, or OXs, with possibly a X or an O at the end of the sequence.
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bump
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File: Automata.png (19 KB, 760x569)
19 KB PNG
>>10468674
S cannot contain a substring of the form XX((OX)^n)X or OO((XO)^n)O
>>
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>>10470376
Use regular expressions to (carefully) uniquely generate all strings that aren't accepted by the DFA
The terms that stop at S are e (the empty string)
... at A are A(BA)* + B(AB)*A
... at B are B(AB)* + A(BA)*B
... at AA are [ A(BA)*A + B(AB)*AA + ( B(AB)*B + A(BA)*BB )(AB)*AA ][ (BA)*BB(AB)*AA ]*
... at BB are [ B(AB)*B + A(BA)*BB + ( A(BA)*A + B(AB)*AA )(BA)*BB ][ (AB)*AA(BA)*BB ]*
... at A_ are [ B(AB)*B + A(BA)*BB + ( A(BA)*A + B(AB)*AA )(BA)*BB ] (AB)*A [ A(BA)*BB(AB)*A ]*
... at B_ are [ A(BA)*A + B(AB)*AA + ( B(AB)*B + A(BA)*BB )(AB)*AA ] (BA)*B [ B(AB)*AA(BA)*B ]*

Converting these to generating functions is pretty easy.
at S ~ 1
at A ~ (t+t^2)/(1-t^2)
at B ~ (t+t^2)/(1-t^2)
at AA ~ [ (t^2 + t^3)/(1-t^2) + (t^2 + t^3)/(1-t^2) * (t^2)/(1-t^2) ] * [ 1/( 1 - (t^4)/( 1-t^2 )^2 ) ]
at BB ~ [ (t^2 + t^3)/(1-t^2) + (t^2 + t^3)/(1-t^2) * (t^2)/(1-t^2) ] * [ 1/( 1 - (t^4)/(1-t^2 )^2 ) ]
at A_ ~ [ (t^2 + t^3)/(1-t^2) + (t^2 + t^3)/(1-t^2) * (t^2)/(1-t^2) ] * (t/(1-t^2)) * [ 1/( 1 - (t^4)/( 1-t^2 )^2 ) ]
at B_ ~ [ (t^2 + t^3)/(1-t^2) + (t^2 + t^3)/(1-t^2) * (t^2)/(1-t^2) ] * (t/(1-t^2)) * [ 1/( 1 - (t^4)/( 1-t^2 )^2 ) ]

Adding all of these terms yields (2t^3 + 2t^2 + 2t + 1))/(1-2t^2)
Which gives (3t + 2)/(1 - 2t^2) - t - 1

If n=0, the coefficient of t^n is 1 (the empty string)
If n=1, the coefficient of t^n is 2
If n>=2 is even, the coefficient of t^n is 2^(n/2 + 1)
If n>=3 is odd, the coefficient of t^n is 3*2^((n-1)/2)

I might be a robot.
>>
I'm gonna save this problem with the intention of solving it when I have time and never look at it again.
>>
>>10470893
I messed up somewhere.
Fuck the regexps.
>>10470376
Just let the nodes represent generating functions and compute them according to which nodes point to a given node.
Let S=1 and A=B, A2 = B2, A_ = B_ by symmetry.
A = St + Bt = t + At so A = t/(1-t)
A2 = At + (A_)t + (B_)t = t(A + 2A_)
A_ = (A2)t

A2 = At + 2(A2)t^2
A2 = At/(1-2t^2) = (t^2)/((1-t)(1-2t^2))

S + A + B + A2 + B2 + A_ + B_ = 1 + 2( t/(1-t) + (t^2 + t^3)/((1-t)(1-2t^2)) )
= (4t + 3)/(1 - 2t^2) - 2/(1-t)

n even gives 3*2^(n/2) - 2
n odd gives 4*2^((n-1)/2) - 2

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text:
# Properties

 Label 64.5.f.a Level $64$ Weight $5$ Character orbit 64.f Analytic conductor $6.616$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 64.f (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.61567763737$$ Analytic rank: $$0$$ Dimension: $$14$$ Relative dimension: $$7$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ Defining polynomial: $$x^{14} - 4 x^{13} + 15 x^{12} - 34 x^{11} + 62 x^{10} - 312 x^{9} + 1432 x^{8} - 4960 x^{7} + 11456 x^{6} - 19968 x^{5} + 31744 x^{4} - 139264 x^{3} + 491520 x^{2} - 1048576 x + 2097152$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{42}$$ Twist minimal: no (minimal twist has level 16) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + \beta_{3} q^{5} + \beta_{9} q^{7} + ( 19 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{12} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + \beta_{3} q^{5} + \beta_{9} q^{7} + ( 19 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{12} ) q^{9} + ( -7 + 7 \beta_{1} - \beta_{7} - \beta_{8} ) q^{11} + ( -1 + \beta_{1} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{9} - \beta_{13} ) q^{13} + ( 24 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{8} - \beta_{10} + 2 \beta_{12} - \beta_{13} ) q^{15} + ( -2 + 2 \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{17} + ( 49 + 49 \beta_{1} + 5 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{19} + ( -13 - 13 \beta_{1} + 15 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{6} + 3 \beta_{9} + \beta_{10} - 3 \beta_{11} - 2 \beta_{12} ) q^{21} + ( -76 + 4 \beta_{2} + 8 \beta_{3} - 4 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} - 8 \beta_{8} - 3 \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{13} ) q^{23} + ( 47 \beta_{1} + 13 \beta_{2} - \beta_{3} - 6 \beta_{4} + 13 \beta_{5} + 3 \beta_{7} - \beta_{8} + \beta_{10} + 3 \beta_{11} + 4 \beta_{12} + \beta_{13} ) q^{25} + ( 115 - 115 \beta_{1} - 2 \beta_{4} + 11 \beta_{5} + 6 \beta_{6} - 3 \beta_{7} + 17 \beta_{8} + 2 \beta_{9} - 6 \beta_{12} + 2 \beta_{13} ) q^{27} + ( 56 - 56 \beta_{1} + 6 \beta_{4} - 36 \beta_{5} + 2 \beta_{6} + 3 \beta_{8} - 6 \beta_{9} - 2 \beta_{12} - 4 \beta_{13} ) q^{29} + ( -20 \beta_{1} - 4 \beta_{2} - 16 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} - 16 \beta_{8} - 3 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} ) q^{31} + ( 16 - 43 \beta_{2} - 4 \beta_{3} + 43 \beta_{5} + 3 \beta_{6} + 4 \beta_{8} - 13 \beta_{9} - 4 \beta_{10} + 4 \beta_{13} ) q^{33} + ( -102 - 102 \beta_{1} + 8 \beta_{2} - 34 \beta_{3} - 6 \beta_{4} - 2 \beta_{6} - 6 \beta_{9} + 6 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} ) q^{35} + ( -115 - 115 \beta_{1} - 59 \beta_{2} - 2 \beta_{3} - 13 \beta_{4} - 8 \beta_{6} - 13 \beta_{9} + 3 \beta_{10} - 5 \beta_{11} - 8 \beta_{12} ) q^{37} + ( -188 - 10 \beta_{2} - 26 \beta_{3} + 10 \beta_{5} - 4 \beta_{6} + 6 \beta_{7} + 26 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} - 6 \beta_{11} + 2 \beta_{13} ) q^{39} + ( 6 \beta_{1} - 67 \beta_{2} - 11 \beta_{3} + 12 \beta_{4} - 67 \beta_{5} + 5 \beta_{7} - 11 \beta_{8} + 3 \beta_{10} + 5 \beta_{11} + 2 \beta_{12} + 3 \beta_{13} ) q^{41} + ( -112 + 112 \beta_{1} + 8 \beta_{4} + 7 \beta_{5} + 4 \beta_{6} - 40 \beta_{8} - 8 \beta_{9} - 4 \beta_{12} + 4 \beta_{13} ) q^{43} + ( 115 - 115 \beta_{1} - 11 \beta_{4} + 129 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + 11 \beta_{9} - 6 \beta_{12} - \beta_{13} ) q^{45} + ( -400 \beta_{1} - 18 \beta_{2} + 34 \beta_{3} - 18 \beta_{5} + 4 \beta_{7} + 34 \beta_{8} + \beta_{10} + 4 \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{47} + ( 15 + 110 \beta_{2} - 16 \beta_{3} - 110 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} + 16 \beta_{8} + 6 \beta_{9} - 4 \beta_{11} ) q^{49} + ( 229 + 229 \beta_{1} + 5 \beta_{2} + 53 \beta_{3} - 12 \beta_{4} - 4 \beta_{6} - 12 \beta_{9} - 4 \beta_{10} - 3 \beta_{11} - 4 \beta_{12} ) q^{51} + ( -65 - 65 \beta_{1} + 187 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} + 2 \beta_{6} + 3 \beta_{9} - 3 \beta_{10} + 9 \beta_{11} + 2 \beta_{12} ) q^{53} + ( 844 - 26 \beta_{2} + 38 \beta_{3} + 26 \beta_{5} + 16 \beta_{6} - 6 \beta_{7} - 38 \beta_{8} + 13 \beta_{9} + 4 \beta_{10} + 6 \beta_{11} - 4 \beta_{13} ) q^{55} + ( -56 \beta_{1} + 165 \beta_{2} - 4 \beta_{3} - \beta_{4} + 165 \beta_{5} - 12 \beta_{7} - 4 \beta_{8} - 4 \beta_{10} - 12 \beta_{11} - 9 \beta_{12} - 4 \beta_{13} ) q^{57} + ( 194 - 194 \beta_{1} + 6 \beta_{4} - 39 \beta_{5} - 22 \beta_{6} + 6 \beta_{7} + 50 \beta_{8} - 6 \beta_{9} + 22 \beta_{12} - 10 \beta_{13} ) q^{59} + ( -303 + 303 \beta_{1} - 5 \beta_{4} - 261 \beta_{5} - 10 \beta_{6} - 7 \beta_{7} - 14 \beta_{8} + 5 \beta_{9} + 10 \beta_{12} + 13 \beta_{13} ) q^{61} + ( 1424 \beta_{1} + 14 \beta_{2} - 30 \beta_{3} + 17 \beta_{4} + 14 \beta_{5} - 12 \beta_{7} - 30 \beta_{8} + 9 \beta_{10} - 12 \beta_{11} + 14 \beta_{12} + 9 \beta_{13} ) q^{63} + ( -108 - 177 \beta_{2} + 31 \beta_{3} + 177 \beta_{5} - 10 \beta_{6} - 7 \beta_{7} - 31 \beta_{8} + 36 \beta_{9} + 17 \beta_{10} + 7 \beta_{11} - 17 \beta_{13} ) q^{65} + ( -573 - 573 \beta_{1} - 50 \beta_{2} - 5 \beta_{3} + 44 \beta_{4} + 16 \beta_{6} + 44 \beta_{9} - 24 \beta_{10} - 5 \beta_{11} + 16 \beta_{12} ) q^{67} + ( 713 + 713 \beta_{1} - 379 \beta_{2} - \beta_{3} + 49 \beta_{4} + 42 \beta_{6} + 49 \beta_{9} - 13 \beta_{10} + 15 \beta_{11} + 42 \beta_{12} ) q^{69} + ( -1484 + 84 \beta_{2} - 8 \beta_{3} - 84 \beta_{5} + 10 \beta_{6} - 30 \beta_{7} + 8 \beta_{8} - 3 \beta_{9} + 7 \beta_{10} + 30 \beta_{11} - 7 \beta_{13} ) q^{71} + ( -242 \beta_{1} - 243 \beta_{2} + 80 \beta_{3} - 41 \beta_{4} - 243 \beta_{5} - 20 \beta_{7} + 80 \beta_{8} - 16 \beta_{10} - 20 \beta_{11} - 17 \beta_{12} - 16 \beta_{13} ) q^{73} + ( -1268 + 1268 \beta_{1} - 58 \beta_{4} - 27 \beta_{5} - 14 \beta_{6} + 12 \beta_{7} - 24 \beta_{8} + 58 \beta_{9} + 14 \beta_{12} - 18 \beta_{13} ) q^{75} + ( -595 + 595 \beta_{1} + 51 \beta_{4} + 359 \beta_{5} - 38 \beta_{6} + 5 \beta_{7} - 3 \beta_{8} - 51 \beta_{9} + 38 \beta_{12} + 17 \beta_{13} ) q^{77} + ( -2172 \beta_{1} + 174 \beta_{2} - 2 \beta_{3} - 20 \beta_{4} + 174 \beta_{5} - 22 \beta_{7} - 2 \beta_{8} + 10 \beta_{10} - 22 \beta_{11} - 28 \beta_{12} + 10 \beta_{13} ) q^{79} + ( 39 + 325 \beta_{2} + 88 \beta_{3} - 325 \beta_{5} - 5 \beta_{6} - 88 \beta_{8} - 37 \beta_{9} + 16 \beta_{10} - 16 \beta_{13} ) q^{81} + ( 1236 + 1236 \beta_{1} - 3 \beta_{2} - 16 \beta_{3} + 30 \beta_{4} - 22 \beta_{6} + 30 \beta_{9} - 14 \beta_{10} - 4 \beta_{11} - 22 \beta_{12} ) q^{83} + ( 649 + 649 \beta_{1} + 377 \beta_{2} - 29 \beta_{3} - 37 \beta_{4} + 28 \beta_{6} - 37 \beta_{9} - \beta_{10} + 15 \beta_{11} + 28 \beta_{12} ) q^{85} + ( 3460 + 84 \beta_{2} - 64 \beta_{3} - 84 \beta_{5} + 42 \beta_{6} - 6 \beta_{7} + 64 \beta_{8} + 5 \beta_{9} - \beta_{10} + 6 \beta_{11} + \beta_{13} ) q^{87} + ( -534 \beta_{1} + 295 \beta_{2} + 44 \beta_{3} + 81 \beta_{4} + 295 \beta_{5} + 44 \beta_{8} + 4 \beta_{10} - 31 \beta_{12} + 4 \beta_{13} ) q^{89} + ( 2044 - 2044 \beta_{1} + 22 \beta_{4} - 30 \beta_{5} - 14 \beta_{6} + 28 \beta_{7} - 136 \beta_{8} - 22 \beta_{9} + 14 \beta_{12} + 14 \beta_{13} ) q^{91} + ( 582 - 582 \beta_{1} - 70 \beta_{4} - 310 \beta_{5} + 20 \beta_{6} + 30 \beta_{7} + 2 \beta_{8} + 70 \beta_{9} - 20 \beta_{12} + 14 \beta_{13} ) q^{93} + ( 3064 \beta_{1} - 20 \beta_{2} + 108 \beta_{3} + 9 \beta_{4} - 20 \beta_{5} + 28 \beta_{7} + 108 \beta_{8} + 16 \beta_{10} + 28 \beta_{11} - 16 \beta_{12} + 16 \beta_{13} ) q^{95} + ( 74 - 248 \beta_{2} - 105 \beta_{3} + 248 \beta_{5} - 15 \beta_{6} + 37 \beta_{7} + 105 \beta_{8} + 23 \beta_{9} - 7 \beta_{10} - 37 \beta_{11} + 7 \beta_{13} ) q^{97} + ( -3450 - 3450 \beta_{1} + 19 \beta_{2} + 106 \beta_{3} - 122 \beta_{4} - 34 \beta_{6} - 122 \beta_{9} - 2 \beta_{10} - 18 \beta_{11} - 34 \beta_{12} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14q + 2q^{3} - 2q^{5} + 4q^{7} + O(q^{10})$$ $$14q + 2q^{3} - 2q^{5} + 4q^{7} - 94q^{11} - 2q^{13} - 4q^{17} + 706q^{19} - 164q^{21} - 1148q^{23} + 1664q^{27} + 862q^{29} - 4q^{33} - 1340q^{35} - 1826q^{37} - 2684q^{39} - 1694q^{43} + 1410q^{45} + 682q^{49} + 3012q^{51} - 482q^{53} + 11780q^{55} + 2786q^{59} - 3778q^{61} - 2020q^{65} - 7998q^{67} + 9628q^{69} - 19964q^{71} - 17570q^{75} - 9508q^{77} + 1454q^{81} + 17282q^{83} + 9948q^{85} + 49284q^{87} + 28036q^{91} + 8896q^{93} - 4q^{97} - 49214q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - 4 x^{13} + 15 x^{12} - 34 x^{11} + 62 x^{10} - 312 x^{9} + 1432 x^{8} - 4960 x^{7} + 11456 x^{6} - 19968 x^{5} + 31744 x^{4} - 139264 x^{3} + 491520 x^{2} - 1048576 x + 2097152$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2875 \nu^{13} + 13444 \nu^{12} - 26581 \nu^{11} + 16062 \nu^{10} - 24954 \nu^{9} + 748984 \nu^{8} - 4619080 \nu^{7} + 10623840 \nu^{6} - 11499840 \nu^{5} + 5960704 \nu^{4} - 51930112 \nu^{3} + 429211648 \nu^{2} - 1316257792 \nu + 1279524864$$$$)/ 678952960$$ $$\beta_{2}$$ $$=$$ $$($$$$-11609 \nu^{13} + 81068 \nu^{12} - 216535 \nu^{11} + 92170 \nu^{10} - 445086 \nu^{9} + 4629992 \nu^{8} - 22287000 \nu^{7} + 63665440 \nu^{6} - 81223104 \nu^{5} + 58993664 \nu^{4} - 414198784 \nu^{3} + 2235252736 \nu^{2} - 5795840000 \nu + 7904428032$$$$)/ 678952960$$ $$\beta_{3}$$ $$=$$ $$($$$$23279 \nu^{13} + 167756 \nu^{12} - 482591 \nu^{11} + 959162 \nu^{10} - 197678 \nu^{9} + 877992 \nu^{8} - 35554520 \nu^{7} + 170800160 \nu^{6} - 398883776 \nu^{5} + 133770240 \nu^{4} - 180005888 \nu^{3} + 2196488192 \nu^{2} - 17134354432 \nu + 45595492352$$$$)/ 678952960$$ $$\beta_{4}$$ $$=$$ $$($$$$-3499 \nu^{13} + 6652 \nu^{12} + 13019 \nu^{11} - 10778 \nu^{10} + 122070 \nu^{9} + 1232776 \nu^{8} - 3340360 \nu^{7} - 34400 \nu^{6} + 24398016 \nu^{5} + 5933568 \nu^{4} - 170023936 \nu^{3} + 637231104 \nu^{2} - 494927872 \nu - 3419275264$$$$)/84869120$$ $$\beta_{5}$$ $$=$$ $$($$$$-30299 \nu^{13} + 175428 \nu^{12} - 396085 \nu^{11} + 537790 \nu^{10} - 1159226 \nu^{9} + 9651512 \nu^{8} - 53625160 \nu^{7} + 150852960 \nu^{6} - 203316544 \nu^{5} + 134759424 \nu^{4} - 982549504 \nu^{3} + 4972855296 \nu^{2} - 16046653440 \nu + 20694433792$$$$)/ 678952960$$ $$\beta_{6}$$ $$=$$ $$($$$$187 \nu^{13} + 588 \nu^{12} - 3179 \nu^{11} + 8306 \nu^{10} - 18854 \nu^{9} + 936 \nu^{8} - 206648 \nu^{7} + 1252384 \nu^{6} - 3000512 \nu^{5} + 5232640 \nu^{4} - 7527424 \nu^{3} + 7045120 \nu^{2} - 93421568 \nu + 359268352$$$$)/2424832$$ $$\beta_{7}$$ $$=$$ $$($$$$54457 \nu^{13} - 666028 \nu^{12} + 2372471 \nu^{11} - 2827082 \nu^{10} - 5318178 \nu^{9} - 5702120 \nu^{8} + 151698840 \nu^{7} - 675003680 \nu^{6} + 1063292352 \nu^{5} + 531149824 \nu^{4} - 2507641856 \nu^{3} - 7324450816 \nu^{2} + 52640448512 \nu - 123104133120$$$$)/ 678952960$$ $$\beta_{8}$$ $$=$$ $$($$$$-63941 \nu^{13} + 248956 \nu^{12} - 325931 \nu^{11} + 303682 \nu^{10} - 1485958 \nu^{9} + 16189512 \nu^{8} - 70629560 \nu^{7} + 157086880 \nu^{6} - 67802816 \nu^{5} + 199142400 \nu^{4} - 922754048 \nu^{3} + 7022559232 \nu^{2} - 16313188352 \nu + 4344512512$$$$)/ 678952960$$ $$\beta_{9}$$ $$=$$ $$($$$$-1373 \nu^{13} - 1028 \nu^{12} + 16237 \nu^{11} - 30894 \nu^{10} + 21354 \nu^{9} + 156488 \nu^{8} + 410632 \nu^{7} - 4973408 \nu^{6} + 13415232 \nu^{5} - 9518080 \nu^{4} - 7715840 \nu^{3} - 46219264 \nu^{2} + 423264256 \nu - 1804861440$$$$)/9699328$$ $$\beta_{10}$$ $$=$$ $$($$$$-60737 \nu^{13} - 163572 \nu^{12} + 819089 \nu^{11} - 871398 \nu^{10} - 2291022 \nu^{9} + 15043880 \nu^{8} + 15561640 \nu^{7} - 288799200 \nu^{6} + 569406528 \nu^{5} + 16526336 \nu^{4} - 1198050304 \nu^{3} + 1135230976 \nu^{2} + 35844947968 \nu - 98412789760$$$$)/ 339476480$$ $$\beta_{11}$$ $$=$$ $$($$$$191167 \nu^{13} - 198708 \nu^{12} - 780399 \nu^{11} + 1949018 \nu^{10} + 2939442 \nu^{9} - 59111000 \nu^{8} + 130720680 \nu^{7} + 55001120 \nu^{6} - 991562688 \nu^{5} + 198482944 \nu^{4} + 4742575104 \nu^{3} - 12649414656 \nu^{2} + 6744604672 \nu + 139540561920$$$$)/ 678952960$$ $$\beta_{12}$$ $$=$$ $$($$$$-107577 \nu^{13} + 484524 \nu^{12} - 1270135 \nu^{11} + 2145610 \nu^{10} - 3926878 \nu^{9} + 32206056 \nu^{8} - 152847000 \nu^{7} + 393864480 \nu^{6} - 547344832 \nu^{5} + 682642432 \nu^{4} - 2131233792 \nu^{3} + 12085280768 \nu^{2} - 41333391360 \nu + 33873985536$$$$)/ 339476480$$ $$\beta_{13}$$ $$=$$ $$($$$$108593 \nu^{13} - 103372 \nu^{12} - 236161 \nu^{11} + 473862 \nu^{10} + 1410158 \nu^{9} - 32112040 \nu^{8} + 64026840 \nu^{7} - 34664480 \nu^{6} - 122103872 \nu^{5} - 344812544 \nu^{4} + 2863123456 \nu^{3} - 10052583424 \nu^{2} + 11285659648 \nu + 31120424960$$$$)/ 339476480$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{12} + \beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{2} - 9 \beta_{1} + 17$$$$)/64$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{12} + 2 \beta_{11} + \beta_{10} - 4 \beta_{9} + 7 \beta_{8} + \beta_{7} - 3 \beta_{6} + 7 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} - 21 \beta_{1} - 63$$$$)/64$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{12} - 2 \beta_{11} + \beta_{10} - 12 \beta_{9} + 31 \beta_{8} + \beta_{7} - 7 \beta_{6} - 65 \beta_{5} - 12 \beta_{4} + 10 \beta_{3} - 20 \beta_{2} + 167 \beta_{1} - 67$$$$)/64$$ $$\nu^{4}$$ $$=$$ $$($$$$-6 \beta_{13} + 11 \beta_{12} + 12 \beta_{11} - 5 \beta_{10} + 12 \beta_{9} + 45 \beta_{8} + 3 \beta_{7} + 29 \beta_{6} - 255 \beta_{5} + 28 \beta_{4} - 4 \beta_{3} + 82 \beta_{2} + 967 \beta_{1} + 121$$$$)/64$$ $$\nu^{5}$$ $$=$$ $$($$$$44 \beta_{13} + 15 \beta_{12} - 34 \beta_{11} - 23 \beta_{10} - 36 \beta_{9} + 107 \beta_{8} + 5 \beta_{7} - 3 \beta_{6} - 29 \beta_{5} + 108 \beta_{4} - 118 \beta_{3} - 4 \beta_{2} - 1909 \beta_{1} + 5561$$$$)/64$$ $$\nu^{6}$$ $$=$$ $$($$$$-38 \beta_{13} - 189 \beta_{12} - 108 \beta_{11} - 5 \beta_{10} - 132 \beta_{9} + 445 \beta_{8} - 157 \beta_{7} + 133 \beta_{6} + 657 \beta_{5} - 68 \beta_{4} - 316 \beta_{3} - 422 \beta_{2} - 8849 \beta_{1} - 12447$$$$)/64$$ $$\nu^{7}$$ $$=$$ $$($$$$-280 \beta_{13} + 7 \beta_{12} + 242 \beta_{11} - 291 \beta_{10} - 84 \beta_{9} + 1143 \beta_{8} + 89 \beta_{7} - 371 \beta_{6} + 1255 \beta_{5} + 156 \beta_{4} - 266 \beta_{3} + 1616 \beta_{2} - 44397 \beta_{1} - 10215$$$$)/64$$ $$\nu^{8}$$ $$=$$ $$($$$$-858 \beta_{13} + 611 \beta_{12} - 1136 \beta_{11} - 153 \beta_{10} - 1156 \beta_{9} + 409 \beta_{8} - 185 \beta_{7} - 867 \beta_{6} - 8315 \beta_{5} - 1284 \beta_{4} + 3976 \beta_{3} + 5494 \beta_{2} - 59217 \beta_{1} + 66329$$$$)/64$$ $$\nu^{9}$$ $$=$$ $$($$$$-2564 \beta_{13} - 153 \beta_{12} + 1158 \beta_{11} - 127 \beta_{10} + 3660 \beta_{9} - 6677 \beta_{8} - 3323 \beta_{7} + 3381 \beta_{6} - 23677 \beta_{5} + 8796 \beta_{4} + 4802 \beta_{3} + 4708 \beta_{2} + 108659 \beta_{1} + 89825$$$$)/64$$ $$\nu^{10}$$ $$=$$ $$($$$$3434 \beta_{13} + 6691 \beta_{12} + 548 \beta_{11} + 2587 \beta_{10} - 4804 \beta_{9} + 3021 \beta_{8} - 1357 \beta_{7} - 5915 \beta_{6} + 53569 \beta_{5} + 21852 \beta_{4} + 7444 \beta_{3} - 121494 \beta_{2} - 580657 \beta_{1} + 628385$$$$)/64$$ $$\nu^{11}$$ $$=$$ $$($$$$-15088 \beta_{13} - 31849 \beta_{12} - 5174 \beta_{11} + 8437 \beta_{10} + 2892 \beta_{9} + 90719 \beta_{8} - 9359 \beta_{7} + 15405 \beta_{6} + 84255 \beta_{5} - 32196 \beta_{4} + 79198 \beta_{3} - 343064 \beta_{2} - 15805 \beta_{1} - 1111399$$$$)/64$$ $$\nu^{12}$$ $$=$$ $$($$$$9886 \beta_{13} + 1059 \beta_{12} + 65352 \beta_{11} - 12833 \beta_{10} + 144956 \beta_{9} + 270289 \beta_{8} + 38927 \beta_{7} - 10163 \beta_{6} - 170835 \beta_{5} + 47548 \beta_{4} + 313424 \beta_{3} + 396334 \beta_{2} - 2474769 \beta_{1} + 1768521$$$$)/64$$ $$\nu^{13}$$ $$=$$ $$($$$$124644 \beta_{13} + 4807 \beta_{12} - 241810 \beta_{11} + 41769 \beta_{10} - 273012 \beta_{9} + 107123 \beta_{8} + 93757 \beta_{7} - 97819 \beta_{6} - 1479205 \beta_{5} + 169308 \beta_{4} + 1013274 \beta_{3} + 1212684 \beta_{2} + 85715 \beta_{1} - 514479$$$$)/64$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/64\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$63$$ $$\chi(n)$$ $$\beta_{1}$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1
 1.03712 − 2.63142i 2.24452 + 1.72109i −2.15805 + 1.82834i −2.40693 − 1.48549i 2.79265 − 0.448449i 0.336831 − 2.80830i 0.153862 + 2.82424i 1.03712 + 2.63142i 2.24452 − 1.72109i −2.15805 − 1.82834i −2.40693 + 1.48549i 2.79265 + 0.448449i 0.336831 + 2.80830i 0.153862 − 2.82424i
0 −11.5209 11.5209i 0 −14.6016 14.6016i 0 24.0210 0 184.461i 0
15.2 0 −5.54016 5.54016i 0 21.7374 + 21.7374i 0 6.62054 0 19.6133i 0
15.3 0 −3.91498 3.91498i 0 4.72348 + 4.72348i 0 −45.3712 0 50.3458i 0
15.4 0 0.0461995 + 0.0461995i 0 −8.04297 8.04297i 0 49.8797 0 80.9957i 0
15.5 0 4.63552 + 4.63552i 0 −29.2002 29.2002i 0 −59.6196 0 38.0239i 0
15.6 0 7.86839 + 7.86839i 0 27.2309 + 27.2309i 0 −50.3097 0 42.8233i 0
15.7 0 9.42589 + 9.42589i 0 −2.84710 2.84710i 0 76.7794 0 96.6949i 0
47.1 0 −11.5209 + 11.5209i 0 −14.6016 + 14.6016i 0 24.0210 0 184.461i 0
47.2 0 −5.54016 + 5.54016i 0 21.7374 21.7374i 0 6.62054 0 19.6133i 0
47.3 0 −3.91498 + 3.91498i 0 4.72348 4.72348i 0 −45.3712 0 50.3458i 0
47.4 0 0.0461995 0.0461995i 0 −8.04297 + 8.04297i 0 49.8797 0 80.9957i 0
47.5 0 4.63552 4.63552i 0 −29.2002 + 29.2002i 0 −59.6196 0 38.0239i 0
47.6 0 7.86839 7.86839i 0 27.2309 27.2309i 0 −50.3097 0 42.8233i 0
47.7 0 9.42589 9.42589i 0 −2.84710 + 2.84710i 0 76.7794 0 96.6949i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 47.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.5.f.a 14
3.b odd 2 1 576.5.m.a 14
4.b odd 2 1 16.5.f.a 14
8.b even 2 1 128.5.f.a 14
8.d odd 2 1 128.5.f.b 14
12.b even 2 1 144.5.m.a 14
16.e even 4 1 16.5.f.a 14
16.e even 4 1 128.5.f.b 14
16.f odd 4 1 inner 64.5.f.a 14
16.f odd 4 1 128.5.f.a 14
48.i odd 4 1 144.5.m.a 14
48.k even 4 1 576.5.m.a 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.5.f.a 14 4.b odd 2 1
16.5.f.a 14 16.e even 4 1
64.5.f.a 14 1.a even 1 1 trivial
64.5.f.a 14 16.f odd 4 1 inner
128.5.f.a 14 8.b even 2 1
128.5.f.a 14 16.f odd 4 1
128.5.f.b 14 8.d odd 2 1
128.5.f.b 14 16.e even 4 1
144.5.m.a 14 12.b even 2 1
144.5.m.a 14 48.i odd 4 1
576.5.m.a 14 3.b odd 2 1
576.5.m.a 14 48.k even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(64, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 2 T + 2 T^{2} - 610 T^{3} - 4613 T^{4} - 31732 T^{5} + 258740 T^{6} - 2114868 T^{7} + 70347753 T^{8} - 127844190 T^{9} + 3102089886 T^{10} - 22201539966 T^{11} + 75486949083 T^{12} - 2442289052376 T^{13} + 20677791784536 T^{14} - 197825413242456 T^{15} + 495269872933563 T^{16} - 11798808601071006 T^{17} + 133534797839563806 T^{18} - 445765127450480190 T^{19} + 19868283272269877193 T^{20} - 48381396305638460148 T^{21} +$$$$47\!\cdots\!40$$$$T^{22} -$$$$47\!\cdots\!72$$$$T^{23} -$$$$56\!\cdots\!13$$$$T^{24} -$$$$60\!\cdots\!10$$$$T^{25} +$$$$15\!\cdots\!22$$$$T^{26} -$$$$12\!\cdots\!82$$$$T^{27} +$$$$52\!\cdots\!21$$$$T^{28}$$
$5$ $$1 + 2 T + 2 T^{2} + 3938 T^{3} - 94565 T^{4} - 9371916 T^{5} - 10800780 T^{6} - 6597908556 T^{7} - 151886931511 T^{8} + 563896366366 T^{9} + 18344031697054 T^{10} - 1149733079121218 T^{11} + 70375486014886875 T^{12} + 1339233603531363800 T^{13} + 14585682673141755608 T^{14} +$$$$83\!\cdots\!00$$$$T^{15} +$$$$27\!\cdots\!75$$$$T^{16} -$$$$28\!\cdots\!50$$$$T^{17} +$$$$27\!\cdots\!50$$$$T^{18} +$$$$53\!\cdots\!50$$$$T^{19} -$$$$90\!\cdots\!75$$$$T^{20} -$$$$24\!\cdots\!00$$$$T^{21} -$$$$25\!\cdots\!00$$$$T^{22} -$$$$13\!\cdots\!00$$$$T^{23} -$$$$86\!\cdots\!25$$$$T^{24} +$$$$22\!\cdots\!50$$$$T^{25} +$$$$71\!\cdots\!50$$$$T^{26} +$$$$44\!\cdots\!50$$$$T^{27} +$$$$13\!\cdots\!25$$$$T^{28}$$
$7$ $$( 1 - 2 T + 8235 T^{2} - 64404 T^{3} + 38860249 T^{4} - 447351454 T^{5} + 125587217723 T^{6} - 1378109878936 T^{7} + 301534909752923 T^{8} - 2578892109370654 T^{9} + 537875867111373049 T^{10} - 2140333660404582804 T^{11} +$$$$65\!\cdots\!35$$$$T^{12} -$$$$38\!\cdots\!02$$$$T^{13} +$$$$45\!\cdots\!01$$$$T^{14} )^{2}$$
$11$ $$1 + 94 T + 4418 T^{2} - 965570 T^{3} - 470088133 T^{4} - 3120961844 T^{5} + 2249641670708 T^{6} + 796542815073292 T^{7} + 97369777194709097 T^{8} - 5317394377195027646 T^{9} -$$$$63\!\cdots\!78$$$$T^{10} -$$$$14\!\cdots\!86$$$$T^{11} -$$$$23\!\cdots\!21$$$$T^{12} +$$$$22\!\cdots\!52$$$$T^{13} +$$$$18\!\cdots\!92$$$$T^{14} +$$$$33\!\cdots\!32$$$$T^{15} -$$$$50\!\cdots\!01$$$$T^{16} -$$$$45\!\cdots\!06$$$$T^{17} -$$$$29\!\cdots\!58$$$$T^{18} -$$$$35\!\cdots\!46$$$$T^{19} +$$$$95\!\cdots\!77$$$$T^{20} +$$$$11\!\cdots\!52$$$$T^{21} +$$$$47\!\cdots\!68$$$$T^{22} -$$$$96\!\cdots\!84$$$$T^{23} -$$$$21\!\cdots\!33$$$$T^{24} -$$$$63\!\cdots\!70$$$$T^{25} +$$$$42\!\cdots\!58$$$$T^{26} +$$$$13\!\cdots\!74$$$$T^{27} +$$$$20\!\cdots\!61$$$$T^{28}$$
$13$ $$1 + 2 T + 2 T^{2} + 6883234 T^{3} + 1464853339 T^{4} + 65707775476 T^{5} + 23817940993652 T^{6} + 16199073445624116 T^{7} + 1142495439970904649 T^{8} +$$$$10\!\cdots\!70$$$$T^{9} +$$$$78\!\cdots\!46$$$$T^{10} +$$$$14\!\cdots\!90$$$$T^{11} +$$$$74\!\cdots\!35$$$$T^{12} +$$$$22\!\cdots\!32$$$$T^{13} +$$$$94\!\cdots\!60$$$$T^{14} +$$$$65\!\cdots\!52$$$$T^{15} +$$$$60\!\cdots\!35$$$$T^{16} +$$$$33\!\cdots\!90$$$$T^{17} +$$$$52\!\cdots\!86$$$$T^{18} +$$$$19\!\cdots\!70$$$$T^{19} +$$$$62\!\cdots\!89$$$$T^{20} +$$$$25\!\cdots\!36$$$$T^{21} +$$$$10\!\cdots\!12$$$$T^{22} +$$$$83\!\cdots\!16$$$$T^{23} +$$$$52\!\cdots\!39$$$$T^{24} +$$$$71\!\cdots\!74$$$$T^{25} +$$$$58\!\cdots\!42$$$$T^{26} +$$$$16\!\cdots\!62$$$$T^{27} +$$$$24\!\cdots\!41$$$$T^{28}$$
$17$ $$( 1 + 2 T + 333755 T^{2} - 12776716 T^{3} + 56419031945 T^{4} - 2961382342882 T^{5} + 6454907058757691 T^{6} - 326099715157486120 T^{7} +$$$$53\!\cdots\!11$$$$T^{8} -$$$$20\!\cdots\!62$$$$T^{9} +$$$$32\!\cdots\!45$$$$T^{10} -$$$$62\!\cdots\!96$$$$T^{11} +$$$$13\!\cdots\!55$$$$T^{12} +$$$$67\!\cdots\!42$$$$T^{13} +$$$$28\!\cdots\!41$$$$T^{14} )^{2}$$
$19$ $$1 - 706 T + 249218 T^{2} - 101381538 T^{3} + 32599242619 T^{4} - 3617133340788 T^{5} - 431513784802892 T^{6} + 960761925731564364 T^{7} -$$$$71\!\cdots\!35$$$$T^{8} +$$$$22\!\cdots\!14$$$$T^{9} -$$$$57\!\cdots\!66$$$$T^{10} +$$$$18\!\cdots\!22$$$$T^{11} -$$$$19\!\cdots\!61$$$$T^{12} -$$$$38\!\cdots\!56$$$$T^{13} +$$$$11\!\cdots\!48$$$$T^{14} -$$$$50\!\cdots\!76$$$$T^{15} -$$$$32\!\cdots\!01$$$$T^{16} +$$$$40\!\cdots\!42$$$$T^{17} -$$$$16\!\cdots\!46$$$$T^{18} +$$$$84\!\cdots\!14$$$$T^{19} -$$$$35\!\cdots\!35$$$$T^{20} +$$$$61\!\cdots\!24$$$$T^{21} -$$$$35\!\cdots\!12$$$$T^{22} -$$$$39\!\cdots\!28$$$$T^{23} +$$$$46\!\cdots\!19$$$$T^{24} -$$$$18\!\cdots\!98$$$$T^{25} +$$$$59\!\cdots\!38$$$$T^{26} -$$$$22\!\cdots\!66$$$$T^{27} +$$$$40\!\cdots\!81$$$$T^{28}$$
$23$ $$( 1 + 574 T + 1024043 T^{2} + 635922028 T^{3} + 551773439769 T^{4} + 314763003369506 T^{5} + 213700110666561659 T^{6} +$$$$10\!\cdots\!08$$$$T^{7} +$$$$59\!\cdots\!19$$$$T^{8} +$$$$24\!\cdots\!86$$$$T^{9} +$$$$12\!\cdots\!49$$$$T^{10} +$$$$38\!\cdots\!08$$$$T^{11} +$$$$17\!\cdots\!43$$$$T^{12} +$$$$27\!\cdots\!34$$$$T^{13} +$$$$13\!\cdots\!81$$$$T^{14} )^{2}$$
$29$ $$1 - 862 T + 371522 T^{2} - 1045006654 T^{3} + 1721779716827 T^{4} - 691721668187596 T^{5} + 502604487474844916 T^{6} -$$$$98\!\cdots\!28$$$$T^{7} +$$$$75\!\cdots\!49$$$$T^{8} -$$$$31\!\cdots\!74$$$$T^{9} +$$$$39\!\cdots\!98$$$$T^{10} -$$$$45\!\cdots\!94$$$$T^{11} +$$$$34\!\cdots\!87$$$$T^{12} -$$$$29\!\cdots\!64$$$$T^{13} +$$$$24\!\cdots\!96$$$$T^{14} -$$$$20\!\cdots\!84$$$$T^{15} +$$$$17\!\cdots\!07$$$$T^{16} -$$$$15\!\cdots\!54$$$$T^{17} +$$$$98\!\cdots\!58$$$$T^{18} -$$$$54\!\cdots\!74$$$$T^{19} +$$$$94\!\cdots\!69$$$$T^{20} -$$$$87\!\cdots\!08$$$$T^{21} +$$$$31\!\cdots\!56$$$$T^{22} -$$$$30\!\cdots\!16$$$$T^{23} +$$$$53\!\cdots\!27$$$$T^{24} -$$$$23\!\cdots\!74$$$$T^{25} +$$$$58\!\cdots\!42$$$$T^{26} -$$$$95\!\cdots\!42$$$$T^{27} +$$$$78\!\cdots\!21$$$$T^{28}$$
$31$ $$1 - 6904334 T^{2} + 24182883262811 T^{4} - 57459081771770667372 T^{6} +$$$$10\!\cdots\!69$$$$T^{8} -$$$$14\!\cdots\!50$$$$T^{10} +$$$$17\!\cdots\!83$$$$T^{12} -$$$$17\!\cdots\!20$$$$T^{14} +$$$$15\!\cdots\!03$$$$T^{16} -$$$$10\!\cdots\!50$$$$T^{18} +$$$$64\!\cdots\!49$$$$T^{20} -$$$$30\!\cdots\!92$$$$T^{22} +$$$$10\!\cdots\!11$$$$T^{24} -$$$$26\!\cdots\!94$$$$T^{26} +$$$$32\!\cdots\!81$$$$T^{28}$$
$37$ $$1 + 1826 T + 1667138 T^{2} + 4976934274 T^{3} + 7030206539163 T^{4} + 1716272691299380 T^{5} + 3798526848883495924 T^{6} +$$$$11\!\cdots\!60$$$$T^{7} -$$$$10\!\cdots\!63$$$$T^{8} -$$$$16\!\cdots\!22$$$$T^{9} +$$$$18\!\cdots\!50$$$$T^{10} -$$$$21\!\cdots\!18$$$$T^{11} -$$$$10\!\cdots\!97$$$$T^{12} -$$$$14\!\cdots\!60$$$$T^{13} -$$$$67\!\cdots\!04$$$$T^{14} -$$$$26\!\cdots\!60$$$$T^{15} -$$$$37\!\cdots\!37$$$$T^{16} -$$$$14\!\cdots\!58$$$$T^{17} +$$$$22\!\cdots\!50$$$$T^{18} -$$$$38\!\cdots\!22$$$$T^{19} -$$$$44\!\cdots\!43$$$$T^{20} +$$$$91\!\cdots\!60$$$$T^{21} +$$$$57\!\cdots\!44$$$$T^{22} +$$$$48\!\cdots\!80$$$$T^{23} +$$$$37\!\cdots\!63$$$$T^{24} +$$$$49\!\cdots\!14$$$$T^{25} +$$$$31\!\cdots\!98$$$$T^{26} +$$$$64\!\cdots\!06$$$$T^{27} +$$$$65\!\cdots\!41$$$$T^{28}$$
$41$ $$1 - 24523982 T^{2} + 302442312166171 T^{4} -$$$$24\!\cdots\!76$$$$T^{6} +$$$$14\!\cdots\!01$$$$T^{8} -$$$$70\!\cdots\!70$$$$T^{10} +$$$$27\!\cdots\!51$$$$T^{12} -$$$$84\!\cdots\!88$$$$T^{14} +$$$$21\!\cdots\!71$$$$T^{16} -$$$$45\!\cdots\!70$$$$T^{18} +$$$$76\!\cdots\!61$$$$T^{20} -$$$$10\!\cdots\!56$$$$T^{22} +$$$$98\!\cdots\!71$$$$T^{24} -$$$$63\!\cdots\!22$$$$T^{26} +$$$$20\!\cdots\!41$$$$T^{28}$$
$43$ $$1 + 1694 T + 1434818 T^{2} + 14278395262 T^{3} + 44454402050619 T^{4} + 13474016894363980 T^{5} + 60977294006539554100 T^{6} +$$$$38\!\cdots\!44$$$$T^{7} +$$$$23\!\cdots\!81$$$$T^{8} -$$$$78\!\cdots\!82$$$$T^{9} +$$$$12\!\cdots\!62$$$$T^{10} +$$$$18\!\cdots\!22$$$$T^{11} -$$$$91\!\cdots\!45$$$$T^{12} -$$$$10\!\cdots\!08$$$$T^{13} +$$$$16\!\cdots\!52$$$$T^{14} -$$$$36\!\cdots\!08$$$$T^{15} -$$$$10\!\cdots\!45$$$$T^{16} +$$$$73\!\cdots\!22$$$$T^{17} +$$$$16\!\cdots\!62$$$$T^{18} -$$$$36\!\cdots\!82$$$$T^{19} +$$$$37\!\cdots\!81$$$$T^{20} +$$$$21\!\cdots\!44$$$$T^{21} +$$$$11\!\cdots\!00$$$$T^{22} +$$$$85\!\cdots\!80$$$$T^{23} +$$$$96\!\cdots\!19$$$$T^{24} +$$$$10\!\cdots\!62$$$$T^{25} +$$$$36\!\cdots\!18$$$$T^{26} +$$$$14\!\cdots\!94$$$$T^{27} +$$$$29\!\cdots\!01$$$$T^{28}$$
$47$ $$1 - 51887758 T^{2} + 1309844227745755 T^{4} -$$$$21\!\cdots\!56$$$$T^{6} +$$$$24\!\cdots\!09$$$$T^{8} -$$$$21\!\cdots\!30$$$$T^{10} +$$$$15\!\cdots\!39$$$$T^{12} -$$$$82\!\cdots\!04$$$$T^{14} +$$$$35\!\cdots\!79$$$$T^{16} -$$$$12\!\cdots\!30$$$$T^{18} +$$$$33\!\cdots\!29$$$$T^{20} -$$$$68\!\cdots\!96$$$$T^{22} +$$$$10\!\cdots\!55$$$$T^{24} -$$$$94\!\cdots\!38$$$$T^{26} +$$$$43\!\cdots\!21$$$$T^{28}$$
$53$ $$1 + 482 T + 116162 T^{2} - 5558326078 T^{3} + 43583027341595 T^{4} + 155411473123116980 T^{5} + 85293132817975457012 T^{6} +$$$$11\!\cdots\!76$$$$T^{7} +$$$$35\!\cdots\!13$$$$T^{8} -$$$$92\!\cdots\!94$$$$T^{9} +$$$$12\!\cdots\!18$$$$T^{10} +$$$$13\!\cdots\!10$$$$T^{11} +$$$$28\!\cdots\!31$$$$T^{12} +$$$$18\!\cdots\!24$$$$T^{13} +$$$$12\!\cdots\!68$$$$T^{14} +$$$$14\!\cdots\!44$$$$T^{15} +$$$$17\!\cdots\!91$$$$T^{16} +$$$$67\!\cdots\!10$$$$T^{17} +$$$$47\!\cdots\!78$$$$T^{18} -$$$$28\!\cdots\!94$$$$T^{19} +$$$$84\!\cdots\!53$$$$T^{20} +$$$$21\!\cdots\!36$$$$T^{21} +$$$$12\!\cdots\!92$$$$T^{22} +$$$$18\!\cdots\!80$$$$T^{23} +$$$$40\!\cdots\!95$$$$T^{24} -$$$$41\!\cdots\!18$$$$T^{25} +$$$$67\!\cdots\!82$$$$T^{26} +$$$$22\!\cdots\!62$$$$T^{27} +$$$$36\!\cdots\!21$$$$T^{28}$$
$59$ $$1 - 2786 T + 3880898 T^{2} - 95235375746 T^{3} + 282835430943931 T^{4} + 951817240129082700 T^{5} +$$$$78\!\cdots\!20$$$$T^{6} +$$$$10\!\cdots\!24$$$$T^{7} -$$$$10\!\cdots\!59$$$$T^{8} +$$$$16\!\cdots\!78$$$$T^{9} +$$$$95\!\cdots\!50$$$$T^{10} +$$$$27\!\cdots\!94$$$$T^{11} +$$$$66\!\cdots\!23$$$$T^{12} -$$$$40\!\cdots\!76$$$$T^{13} +$$$$34\!\cdots\!76$$$$T^{14} -$$$$49\!\cdots\!36$$$$T^{15} +$$$$97\!\cdots\!83$$$$T^{16} +$$$$49\!\cdots\!14$$$$T^{17} +$$$$20\!\cdots\!50$$$$T^{18} +$$$$42\!\cdots\!78$$$$T^{19} -$$$$34\!\cdots\!99$$$$T^{20} +$$$$39\!\cdots\!04$$$$T^{21} +$$$$36\!\cdots\!20$$$$T^{22} +$$$$53\!\cdots\!00$$$$T^{23} +$$$$19\!\cdots\!31$$$$T^{24} -$$$$78\!\cdots\!06$$$$T^{25} +$$$$38\!\cdots\!58$$$$T^{26} -$$$$33\!\cdots\!66$$$$T^{27} +$$$$14\!\cdots\!41$$$$T^{28}$$
$61$ $$1 + 3778 T + 7136642 T^{2} + 13584988130 T^{3} + 15885658886619 T^{4} - 322904339201283724 T^{5} -$$$$12\!\cdots\!20$$$$T^{6} -$$$$11\!\cdots\!80$$$$T^{7} -$$$$34\!\cdots\!59$$$$T^{8} -$$$$57\!\cdots\!82$$$$T^{9} -$$$$60\!\cdots\!50$$$$T^{10} -$$$$13\!\cdots\!70$$$$T^{11} +$$$$15\!\cdots\!55$$$$T^{12} +$$$$19\!\cdots\!60$$$$T^{13} +$$$$92\!\cdots\!08$$$$T^{14} +$$$$26\!\cdots\!60$$$$T^{15} +$$$$29\!\cdots\!55$$$$T^{16} -$$$$35\!\cdots\!70$$$$T^{17} -$$$$22\!\cdots\!50$$$$T^{18} -$$$$29\!\cdots\!82$$$$T^{19} -$$$$24\!\cdots\!19$$$$T^{20} -$$$$11\!\cdots\!80$$$$T^{21} -$$$$16\!\cdots\!20$$$$T^{22} -$$$$60\!\cdots\!64$$$$T^{23} +$$$$41\!\cdots\!19$$$$T^{24} +$$$$48\!\cdots\!30$$$$T^{25} +$$$$35\!\cdots\!02$$$$T^{26} +$$$$25\!\cdots\!38$$$$T^{27} +$$$$95\!\cdots\!61$$$$T^{28}$$
$67$ $$1 + 7998 T + 31984002 T^{2} + 47670849246 T^{3} - 439076236005637 T^{4} - 648765759780793716 T^{5} +$$$$99\!\cdots\!64$$$$T^{6} +$$$$93\!\cdots\!52$$$$T^{7} +$$$$28\!\cdots\!21$$$$T^{8} -$$$$69\!\cdots\!70$$$$T^{9} -$$$$55\!\cdots\!14$$$$T^{10} -$$$$21\!\cdots\!74$$$$T^{11} -$$$$39\!\cdots\!49$$$$T^{12} -$$$$17\!\cdots\!60$$$$T^{13} -$$$$39\!\cdots\!76$$$$T^{14} -$$$$35\!\cdots\!60$$$$T^{15} -$$$$15\!\cdots\!09$$$$T^{16} -$$$$17\!\cdots\!14$$$$T^{17} -$$$$92\!\cdots\!34$$$$T^{18} -$$$$23\!\cdots\!70$$$$T^{19} +$$$$18\!\cdots\!41$$$$T^{20} +$$$$12\!\cdots\!32$$$$T^{21} +$$$$27\!\cdots\!04$$$$T^{22} -$$$$35\!\cdots\!96$$$$T^{23} -$$$$48\!\cdots\!37$$$$T^{24} +$$$$10\!\cdots\!66$$$$T^{25} +$$$$14\!\cdots\!82$$$$T^{26} +$$$$72\!\cdots\!78$$$$T^{27} +$$$$18\!\cdots\!81$$$$T^{28}$$
$71$ $$( 1 + 9982 T + 145017323 T^{2} + 1165310044396 T^{3} + 10014081489420185 T^{4} + 63641985337531169890 T^{5} +$$$$40\!\cdots\!59$$$$T^{6} +$$$$20\!\cdots\!72$$$$T^{7} +$$$$10\!\cdots\!79$$$$T^{8} +$$$$41\!\cdots\!90$$$$T^{9} +$$$$16\!\cdots\!85$$$$T^{10} +$$$$48\!\cdots\!16$$$$T^{11} +$$$$15\!\cdots\!23$$$$T^{12} +$$$$26\!\cdots\!42$$$$T^{13} +$$$$68\!\cdots\!61$$$$T^{14} )^{2}$$
$73$ $$1 - 168573838 T^{2} + 13353714116727067 T^{4} -$$$$70\!\cdots\!96$$$$T^{6} +$$$$30\!\cdots\!57$$$$T^{8} -$$$$11\!\cdots\!98$$$$T^{10} +$$$$39\!\cdots\!07$$$$T^{12} -$$$$11\!\cdots\!44$$$$T^{14} +$$$$31\!\cdots\!67$$$$T^{16} -$$$$74\!\cdots\!78$$$$T^{18} +$$$$15\!\cdots\!37$$$$T^{20} -$$$$29\!\cdots\!16$$$$T^{22} +$$$$45\!\cdots\!67$$$$T^{24} -$$$$46\!\cdots\!78$$$$T^{26} +$$$$22\!\cdots\!61$$$$T^{28}$$
$79$ $$1 - 364033678 T^{2} + 65454647116587227 T^{4} -$$$$76\!\cdots\!56$$$$T^{6} +$$$$65\!\cdots\!81$$$$T^{8} -$$$$43\!\cdots\!10$$$$T^{10} +$$$$23\!\cdots\!67$$$$T^{12} -$$$$99\!\cdots\!88$$$$T^{14} +$$$$35\!\cdots\!87$$$$T^{16} -$$$$10\!\cdots\!10$$$$T^{18} +$$$$23\!\cdots\!61$$$$T^{20} -$$$$40\!\cdots\!96$$$$T^{22} +$$$$52\!\cdots\!27$$$$T^{24} -$$$$44\!\cdots\!58$$$$T^{26} +$$$$18\!\cdots\!21$$$$T^{28}$$
$83$ $$1 - 17282 T + 149333762 T^{2} - 1088719641698 T^{3} + 16964332297412731 T^{4} -$$$$21\!\cdots\!96$$$$T^{5} +$$$$17\!\cdots\!52$$$$T^{6} -$$$$12\!\cdots\!12$$$$T^{7} +$$$$11\!\cdots\!61$$$$T^{8} -$$$$11\!\cdots\!62$$$$T^{9} +$$$$90\!\cdots\!74$$$$T^{10} -$$$$61\!\cdots\!82$$$$T^{11} +$$$$44\!\cdots\!67$$$$T^{12} -$$$$35\!\cdots\!48$$$$T^{13} +$$$$26\!\cdots\!76$$$$T^{14} -$$$$16\!\cdots\!08$$$$T^{15} +$$$$10\!\cdots\!47$$$$T^{16} -$$$$65\!\cdots\!02$$$$T^{17} +$$$$45\!\cdots\!94$$$$T^{18} -$$$$28\!\cdots\!62$$$$T^{19} +$$$$13\!\cdots\!81$$$$T^{20} -$$$$66\!\cdots\!92$$$$T^{21} +$$$$44\!\cdots\!72$$$$T^{22} -$$$$26\!\cdots\!76$$$$T^{23} +$$$$98\!\cdots\!31$$$$T^{24} -$$$$29\!\cdots\!58$$$$T^{25} +$$$$19\!\cdots\!42$$$$T^{26} -$$$$10\!\cdots\!02$$$$T^{27} +$$$$29\!\cdots\!81$$$$T^{28}$$
$89$ $$1 - 548528910 T^{2} + 149200943223060123 T^{4} -$$$$26\!\cdots\!16$$$$T^{6} +$$$$35\!\cdots\!49$$$$T^{8} -$$$$37\!\cdots\!50$$$$T^{10} +$$$$31\!\cdots\!11$$$$T^{12} -$$$$21\!\cdots\!00$$$$T^{14} +$$$$12\!\cdots\!91$$$$T^{16} -$$$$57\!\cdots\!50$$$$T^{18} +$$$$21\!\cdots\!09$$$$T^{20} -$$$$64\!\cdots\!36$$$$T^{22} +$$$$14\!\cdots\!23$$$$T^{24} -$$$$20\!\cdots\!10$$$$T^{26} +$$$$14\!\cdots\!61$$$$T^{28}$$
$97$ $$( 1 + 2 T + 387850619 T^{2} + 251760181236 T^{3} + 75114732161345545 T^{4} + 73269666487293981214 T^{5} +$$$$94\!\cdots\!67$$$$T^{6} +$$$$89\!\cdots\!44$$$$T^{7} +$$$$83\!\cdots\!27$$$$T^{8} +$$$$57\!\cdots\!54$$$$T^{9} +$$$$52\!\cdots\!45$$$$T^{10} +$$$$15\!\cdots\!56$$$$T^{11} +$$$$21\!\cdots\!19$$$$T^{12} +$$$$96\!\cdots\!62$$$$T^{13} +$$$$42\!\cdots\!61$$$$T^{14} )^{2}$$
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text:
# How to create multi-row table?

I want to draw the following table :

I have used the following latex code to do the above table :

\documentclass[12pt,twoside,a4paper]{book}
\usepackage[left=.45in,right=.45in,top=.6in,bottom=.6in,headheight=14.5pt]{geometry}
\usepackage{multirow}
\begin{document}

\begin{tabular}{|c|c|c|c|c|c|c|}\hline
{\bf  Prefix} & {\bf  Type } & {\bf  No of digits} & {\bf  2nd  digits} & {\bf  Text number} & {\bf   Text-number} & {\bf  Text digit} \\ \hline
\multicolumn{2}{|c|}{\multirow{2}{*}{12-14}} & Text-MCCC\\ \cline{3-3}
\multicolumn{2}{|c|}{} & Text-VSSS \\\hline
\end{tabular}

\end{document}

This produces :

Anyone can tell me how can I draw my desired table. Thanks

• Don't use \bf .. it's deprecated. Use \textbf{...} or (since it's inside a tabular cell) \bfseries – user31729 Feb 15 '15 at 11:15
• I am not sure you need \multicolumn at all – user31729 Feb 15 '15 at 11:25
• You don't have any cells spanning more than 1 column in your sample image. – David Carlisle Feb 15 '15 at 12:07
• What cells/columns should appear as paragraphs is not clear to me. – Bernard Feb 15 '15 at 12:08

## 2 Answers

This should let you start. The missing rows are left as an exercise for you.

\documentclass[12pt,twoside,a4paper]{book}
\usepackage[left=.45in,right=.45in,top=.6in,bottom=.6in,headheight=14.5pt]{geometry}
\usepackage{fmtcount} % for \ordinalnum
\usepackage{array,multirow}

\newcolumntype{C}[1]{>{\centering\arraybackslash}m{#1}}

\begin{document}

\begin{tabular}{|c|c|C{1.2cm}|C{1.2cm}|c|c|C{1.2cm}|}
\hline
\textbf{Prefix} & \textbf{Type} & \textbf{No of digits} & \textbf{\ordinalnum{2} digits}
& \textbf{Text number} & \textbf{Text-number} & \textbf{Text digit} \\
\hline
\multirow{4}{*}{12-14} & \multirow{4}{*}{Text-MCCC} & \multirow{4}{*}{16} & 1
& \ordinalnum{2}-\ordinalnum{3} & \multirow{4}{*}{Long text here} & \multirow{4}{1.2cm}{\centering Text with digits}\\
\cline{4-5}
& & & 2 & \ordinalnum{2}-\ordinalnum{4} & & \\
\cline{4-5}
& & & 3 & \ordinalnum{2}-\ordinalnum{5} & & \\
\cline{4-5}
& & & 4 & \ordinalnum{2}-\ordinalnum{6} & & \\
\hline
\end{tabular}

\end{document}

I really think that creating such tables is much easier in ConTeXt than in LaTeX. For example, this is how you can create such a table using Natural Tables (which work quite similar to HTML tables). Note that the syntax is readable and there is clean separation of content and presentation.

\startsetups table:style
\setupTABLE[row][1][style=bold]
\setupTABLE[each][each][align={middle,lohi}, offset=0.5em]
\setupTABLE[column][last][width=1.5cm]
\stopsetups
\starttext

\bTABLE[setups={table:style}]
\bTR
\bTD Prefix \eTD
\bTD Type   \eTD
\bTD No of \\ digits \eTD
\bTD 2nd \\ digits   \eTD
\bTD Text number     \eTD
\bTD Text-number     \eTD
\bTD Text \\ digit   \eTD
\eTR
\bTR
\bTD[nr=4] 12-14     \eTD
\bTD[nr=4] Text-MCCC \eTD
\bTD[nr=4] 16        \eTD
\bTD 1               \eTD
\bTD 2nd--3rd        \eTD
\bTD[nr=4] Long text here \eTD
\bTD[nr=4] Text with digits \eTD
\eTR
\bTR
\bTD 2        \eTD
\bTD 2nd--4th \eTD
\eTR
\bTR
\bTD 3        \eTD
\bTD 2nd--5th \eTD
\eTR
\bTR
\bTD 4        \eTD
\bTD 2nd--6th \eTD
\eTR
\bTR
\bTD[nr=2]  4        \eTD
\bTD[nr=2] Text-VSSS \eTD
\bTD 13              \eTD
\bTD --              \eTD
\bTD[nr=2] 2nd--6th  \eTD
\bTD 7th--12th       \eTD
\bTD Text with digits\eTD
\eTR
\bTR
\bTD 16         \eTD
\bTD --         \eTD
\bTD 7th--15th  \eTD
\bTD Text with digits \eTD
\eTR
\eTABLE

\stoptext

which gives