Re: From the LMFDB feedback page

[John Jones]

I think I know the answers to your questions, but am forwarding this to the LMFDB support list first to see if someone more familiar with the theory can answer it.

John Jones

On Mon, Apr 18, 2022 at 1:09 PM Hrvoje Abraham <ahr...@gmail.com> wrote:

Hello,

Thank you for your prompt and clear answer.

I assumed this is so, but was not able to verify it numerically as everything was mismatched, so I suspected I misunderstood the principle. I evaluated the function via q expansion all up to order 100, did not work, that at some point it succeeded with 100,000 terms of infinite product and you really need so many to reach the precision, and finally simply used eta relation as eta(z) is available in some numerical libraries, and it works perfectly.

Then, I noticed some are not available via eta quotient. Is there some theorem for specific N, k, chi for which this is not possible, or it was simply not yet found or not documented? Or some other smarter hint how to evaluate them numerically to get "numerical modularity"? E.g.:

https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/19/2/a/a/

Hrvoje

On Mon, Apr 18, 2022 at 9:36 PM John Jones <j...@asu.edu> wrote:

Hi,

On the LMFDB feedback page, you wrote:

Hello, Thank you for creating this impressive site of the ultimate math beauty. I'm a layman compared to pro maths, do some research & numerical experiments for hobby, and I wonder for some modular form, e.g. link provided, where can I get or identify underlying SL linear transformation matrix, so (a b; c d)? Thank you in advance, Hrvoje

https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/12/12/a/a/

I am not sure what you mean.  The form satisfies an equation for every matrix M in a certain subgroup of SL(2,Z) (see https://www.lmfdb.org/knowledge/show/cmf?timestamp=1545201145469776).  In this case, the subgroup is Gamma_0(12) because it is level 12 (see https://www.lmfdb.org/knowledge/show/group.sl2z.subgroup.gamma0n?timestamp=1555023477327750 ).  This is an infinite group, so there are infinitely many matrices involved.

I hope that answers your question.

John Jones

[David Farmer]

There are only finitely many modular forms which are eta quotients.
This paper might give the full answer:

https://arxiv.org/abs/1311.1460

If you are checking a modular relationship numerically, it is necessary
to stay close to the cusp where you have done the Fourier expansion.

If the previous sentence is not clear, then it will be helpful to
consult a basic reference on Modular forms, such as

Topics in Classical Automorphic Forms, by Henryk Iwaniec

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[Andrew Sutherland]

On the CMF completeness page

    https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/Completeness

the last bullet item notes that the database includes all 43 newforms that can be expressed as eta quotients and provides a reference to the literature (https://www.ams.org/journals/tran/1996-348-12/S0002-9947-96-01743-6/) where this is proved.

[Hrvoje Abraham]

Hello,

Thank you for your clear answer and the paper.

Regarding numerics, I empirically detected an area which demonstrates modularity numerically, and understand the reasoning and importance of staying close to cusp for q-expansion, it is to be seen how to proceed to widen that domain and make more robust calculation.

Just a small final question, does LMFDB present eta quotient for all forms it can do so, or only for a subset?

Hrvoje

[David Farmer]

Did you see Drew Sutherland's reply, citing this:

https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/Completeness

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[Hrvoje Abraham]

Hello,

No, it did not reach me for some reason. I just requested group access.

This is great, so there are only a handful of these eta-jewels, very nice! :)

Thank you once more for some beautiful math,

Hrvoje