Notes from meeting of Maass forms working group 31.07.07:
What data exists (online)?
- Booker-Strömbergsson:
- First 2000 eigenvalues on PSL(2,Z) (rigorously computed)
- Small eigenvalues on Gamma_0(N) with Dirichlet character and N large?.
Online: Some ev/coeff to high accuracy on PSL(2,Z): http://www.math.uu.se/~astrombe/emaass/emaass.html
- Farmer-Lemurell: Lots of eigenvalues on Gamma_0(N) for small to moderate N.
Online: nothing yet.
- Hejhal: Eigenvalues on PSL(2,Z) and Hecke triangle groups G_q (q=4,5,6,7 at least).
- Strömberg:
- Eigenvalues on Gamma_0(N) for N up to 109. With Dirichlet character.
- Larger sets for certain N, e.g. 5.
- Fourier coefficients of forms on e.g. N=5 with the quadratic Dirichlet character.
- Eigenvalues and Fourier coefficients on PSL(2,Z) with arbitrary real weight and eta multiplier.
- Eigenvalues/coefficients for Gamma_0(4) weight 1/2, 3/2 with theta multiplier.
Online: Miscellaneous data/pictures of Maass waveforms. See e.g. http://www.dynamik.tu-clausthal.de/~fredrik/research/
What algorithms exist (available)?
There are two "main" algorithms for finding Maass waveforms * the "automorphy" algorithm in the version by Hejhal generalized by Strömberg to subgroups of the modular group and arbitrary weight and multiplier (character). This algorithm also produces Fourier coefficients. Implementation in Fortran 95 and not (yet) available.
- another kind of automorphy algorithm by Farmer-Lemurell. Implementation in Mathematika and not available.
- trace formula algorithm by Booker-Strömbergsson. Implemented in Pari and not available.
Todo:
All persons with data should first of all put up whatever they already have independent of "niceness" of data. If you are not comfortable with the asserted accuracy just put up a disclaimer that the data is in "sandbox" format. Present data in ascii format, machine friendly. Algorithms should be made available too.
Notes from meeting of Maass forms working group 02.08.07:
* Q: What data should be attached to a Maass form? * A: Level, Weight, Multiplier/Character, Laplace eigenvalue, even/odd with respect to reflection. Even/odd wrt to all Atkin-Lehner involutons. Namingschemes for characters needs to be described in corresponding README files. Everyone involved agreed to start uploading data as soon as possible.