q-expansions of normalized eigenforms of level 1

[Kapil Chandran]

Dear LMFDB Editorial Board, 

 

Thank you so much for organizing this website! 

We are currently working on a research paper on partition functions and are in need of the coefficients of the normalized eigenforms for the space of cusp forms S_{12k} of level 1 for small values of k. Specifically, we are interested in the determinant of the d x d matrix consisting of the first d coefficients of each of the d normalized eigenforms, where d = dim S_{12k}.

On the website, we've found q-expansions for newforms, but we are unsure how to find q-expansions for eigenforms using the available search functionality. We were wondering if you could assist us in using the search feature of your website, or possibly direct us to any code you're aware of which could help us calculate these coefficients if they are not available on LMFDB.

Thanks,

Kapil, Erin, and Yunseo 

[David Farmer]

In level 1 everything is new, so newforms and normalized eigenforms
are the same thing.

Does that address your question, or was there something else?

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[Kapil Chandran]

Hi David,

Thank you for the explanation. I am still a bit confused. For example, I would like to find the q-expansions of the 3 normalized eigenforms for S_{36}(Gamma_0(1)). When I search for newforms of level 1 and weight 36 on LMFDB, I only see one search result (labeled 1.36.a.a). Are you saying that this search result is already a normalized eigenform? Also, how would I go about finding the other 2 normalized eigenforms using the search feature? Thank you for your help.

Best,

Kapil

[David Farmer]

Dear Kapil,

The vector space S_{36}(Gamma_0(1)) is 3-dimensional. The three newforms
are Galois conjugates of each other and the coefficients can be expressed
in terms of the roots of one degree 3 polynomial (which happens to be
x^3 - 12422194 x - 2645665785 in this example).

On this page

https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/1/36/a/a/

a root of that polynomial is \nu, and \beta_j for j = 0, 1, 2 are
expressed in terms of \nu. Then the newform f has a q-expansion in
terms of the \beta_j. So really there are 3 newforms, one for
each value of \nu.

On that same page, those three newforms are called "embeddings".

Each embedding has a separate home page, but on that page the coefficients
are written as decimals.

Maeda's conjecture says that for level 1, for each weight the newforms
can always be expressed in terms of the roots of a single irreducible
polynomial. Everyone believes that conjecture, and it is true in every
example that has been observed. So, the situation in weight 36 is
typical.

Regards,

David

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[Edgar Costa]

The dimension of S_{36}(Gamma_0(1))  as complex vector space is 3 as you noticed.

If you click on the link for 1.36.a.a https://beta.lmfdb.org/ModularForm/GL2/Q/holomorphic/1/36/a/a/ you will notice that the eigenform is defined over a cubic field, and thus if you consider the 3 possible embeddings of that cubic field to the complex numbers you get what we call embedded eigenforms:

https://beta.lmfdb.org/ModularForm/GL2/Q/holomorphic/1/36/a/a/1/1/

https://beta.lmfdb.org/ModularForm/GL2/Q/holomorphic/1/36/a/a/1/2/

https://beta.lmfdb.org/ModularForm/GL2/Q/holomorphic/1/36/a/a/1/3/

you would be able to find links to these embedded newforms under the section "Embeddings".

I assume you are interested in downloading the q-expansions for these embedded eigenforms, to do this automatically I recommend using the API:

https://beta.lmfdb.org/api/mf_hecke_cc/?level=1&weight=36

you may also obtain it in json/yaml format.

For example, if you want all the embedded eigenforms of level 1 in json format, you will  have to stitch these 20 jsons:

https://beta.lmfdb.org/api/mf_hecke_cc/?_format=json&level=1&_offset=0

https://beta.lmfdb.org/api/mf_hecke_cc/?_format=json&level=1&_offset=100

...

https://beta.lmfdb.org/api/mf_hecke_cc/?_format=json&level=1&_offset=2000

as we return data in blocks of 100 results.

Best,

Edgar

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[Kapil Chandran]

Dear Edgar and David,

Thank you very much for your explanations and help! I was able to download the Fourier coefficients to Sage and compute the quantities I was interested in.

Best,

Kapil

[cafe wilson]

sko wooks