### Description

Going to Airflow web UI, Browse>DAG Run displays the list of runs, but there is no way to search or filter based on the text in the "Note" column.

### Use case/motivation

It is possible to do a free text search for the "Run Id" field. The Note field may contain pieces of information that may be relevant to find, or to filter on the basis of these notes.

### Related issues

Sorting by Note in List Dag Run fails: 
https://github.com/apache/airflow/issues/30041

### Are you willing to submit a PR?

- [ ] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

dag_run: DagRun | None, *, json_encoder: type[json.JSONEncoder] = json.JSONEncoder
) -> dict[str, Any] | None:
    if not dag_run:
        return None
    conf, conf_is_json = get_dag_run_conf(dag_run.conf, json_encoder=json_encoder)
    return {
        "run_id": dag_run.run_id,
        "queued_at": datetime_to_string(dag_run.queued_at),
        "start_date": datetime_to_string(dag_run.start_date),
        "end_date": datetime_to_string(dag_run.end_date),
        "state": dag_run.state,
        "execution_date": datetime_to_string(dag_run.execution_date),
        "data_interval_start": datetime_to_string(dag_run.data_interval_start),
        "data_interval_end": datetime_to_string(dag_run.data_interval_end),
        "run_type": dag_run.run_type,
        "last_scheduling_decision": datetime_to_string(dag_run.last_scheduling_decision),
        "external_trigger": dag_run.external_trigger,
        "conf": conf,
        "conf_is_json": conf_is_json,
        "note": dag_run.note,
    }
def check_import_errors(fileloc, session):
    
    import_errors = session.query(errors.ImportError).filter(errors.ImportError.filename == fileloc).all()
    if import_errors:
        for import_error in import_errors:
            flash("Broken DAG: [{ie.filename}] {ie.stacktrace}".format(ie=import_error), "dag_import_error")
def check_dag_warnings(dag_id, session):