Petersson norm of Maass cusp forms

[Nathan Benjamin]

Dear LMFDB team,

Is there a way to lookup (and/or calculate) high-precision values of the Petersson norm of SL(2,Z) Maass cusp forms? In other words, I want to numerically compute the integral $\int_{-1/2}^{1/2} dx \int_{\sqrt{1-x^2}}^{\infty} dy |\nu(\tau)|^2 y^{-2}$, where $\nu(\tau)$ is a SL(2,Z) Maass cusp form. For instance the first cusp form with spectral parameter R = 9.533... has Petersson norm about 1.679 * 10^{-14}. 

Is this data either explicitly available or easily calculable from known data? I know it can be calculated by

<a^2> * \pi^2/(48 \cosh(\pi R))

where a are the Fourier coefficients of the cusp form (i.e. the Hecke eigenvalues) and R the spectral parameter. However this formula seems only numerically accurate to around ~3 decimals due to only ~1000 available a's for low-lying cusp forms. 

Is a much higher-precision estimate (e.g. ~10 decimal places) for the Petersson norm either (a) available on the database, or (b) easily calculable from data available (e.g. using some other identities I'm not aware of)? 

Thank you for your time and help!

Best,
Nathan

[David Lowry-Duda]

Dear Nathan Benjamin,

Unfortunately, we haven't computed the Peterrsson norms for the Maass forms. I haven't thought about how to do this rigorously, and for higher level Maass forms we would need to be able to certify forms to higher precision in order to say something meaningful about the Petersson norms.

I have an enormous amount more data for level 1 forms that is not in the LMFDB, though. I'd be happy to send you as many coefficients as you'd like, within reasonable limits.

On 15:36 Mon 11 Aug 2025, Nathan Benjamin wrote:
>I know it can be calculated by
>
><a^2> * \pi^2/(48 \cosh(\pi R))
>
>where a are the Fourier coefficients of the cusp form (i.e. the Hecke
>eigenvalues) and R the spectral parameter.

Alternately, I could compute the SL2Z Petersson norms from the data. As the data can be large, this might be convenient. But I don't quite understand the method of computation you were referring to just above. Could you say a bit more about what you mean?

- DLD

[Nathan Benjamin]

Dear David,

Thank you so much for your response! 

By <a^2> * \pi^2/(48 \cosh(\pi R)):

<a^2> is the average of the Hecke eigenvalues^2. For example, for the first cusp form, which has spectral parameter R=9.534, it would be the average of: (1)^2, (-1.068)^2, (-0.456)^2, (0.141)^2, etc. I think this can be derived by considering the triple integral (Eisenstein * cusp^2) and using the Rankin-Selberg unfolding trick on the Eisenstein, and then taking the residue as s approaches 1. If you know of a better way to get this numerically (I am only interested in SL2Z, which I think is level 1), I would be very interested!

If it is not too much trouble, I would be extremely grateful if you could send me the Petersson norms based on the formula above using your internal data. The first e.g. 30 or so cusp forms would be fine, if that is easy for you to compute. Thank you so much for your help! 

Best

Nathan