Maass eigenvalues

[jpan]

Dear Sir or Madam,

Is your list of the first 69 Maass eigenvalues of level 1, weight 0
(https://www.lmfdb.org/ModularForm/GL2/Q/Maass/?search_type=List&all=1)
complete? I ask this because in Peter Sarnak's paper in 2003
(https://www.ams.org/journals/bull/2003-40-04/S0273-0979-03-00991-1/S0273-0979-03-00991-1.pdf),
he gave a figure of the 33rd Maass form (Figure 3, page 445), whereas he
claimed the corresponding eigenvalue is 916.52, a value different from
your 33rd eigenvalue (31.5265^2+0.25 = 994.23...). He accredited others'
reference for the graph, but the referred preprint does not have
publication record on mathsci.

Thank you for your time.

Jiakun Pan

[David Farmer]

Dear Jiakun Pan,

You are correct that there are missing Maass forms in the LMFDB.

In fact, I am pretty sure that for level 1, the first 31 or 32
Maass forms are in the database (and correct). Immediately after
that, some are missing (I suspect more than one is missing),
and once the list resumes, there are sporadic omissions throughout
the displayed range.

We are in the midst of a long-term project to recompute the Maass forms
and also to certify that the list is complete and the displayed
decimal places are correct. It is possible (if things go well) that
the new data for level 1 will be available by the end of the year,
and prime level a short time later.

The LMFDB documentation indicates that we do not claim the lists to
be complete (because the previous methods are unable to check
completeness, but the new methods can check that), but I will
update the documentation to give details about the level 1 case.

Please do not hesitate to contact us again if you have any
questions or notice additional discrepancies.

Regards,

David Farmer

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