# Hejhal's algorithm and computational methods for non-classical Maass wave forms

Hejhal's algorithm [1] was a little gadget invented in the 90's for calculating the Hecke eigenvalues and Fourier coefficients of Maass wave forms. Later, Booker, Strombergsson, and Venkatesh (BSV) [2] took Hejhal's paper and made it more viable to perform high-precision calculations.

The basic idea is to assume the Fourier expansion takes a certain form, and write down a finite linear system (where the finitude is possible due to assumptions on the final order of magnitude of desired precision). Now take a horocycle on the complex upper half plane, and use these points in the linear system, and you can almost solve it. In BSV's words, "to solve for r [the desired eigenvalue] the above linear system is repeatedly solved for two different Y-values [horocycles], successively adjusting r to make the two solution vectors as nearly equal as possible."

My question is whether such an algorithm or the germ of an idea for such an algorithm exists for non-classical Maass wave forms. What would take the analogue of horocycles, and would we have enough information to solve an analogous linear system? (Remember, Fourier expansions are a good deal more complicated in multiple dimensions)

[1] Dennis A. Hejhal. On eigenfunctions of the Laplacian for Hecke triangle groups. In Emerging applications of number theory (Minneapolis, MN, 1996), volume 109 of IMA Vol. Math. Appl., pages 291–315. Springer, New York, 1999.

• The second reference should be Booker, Strömbergsson and Venkatesh, not just Venkatesh Commented Apr 14, 2013 at 19:20
• Thanks, I got blindsighted by the fact it was hosted on his website. Commented Apr 14, 2013 at 20:07
• What is an analogue of a Hecke eigenvalue, what is the Fourierexpansion, how to prove existence of cuspidal Maass forms....i do not expect something useful can be said in this generality. Commented Apr 14, 2013 at 20:37
• Are you asking for higher rank automorphic forms or non-congruence stuff. My previous comment applies to the later only. Please specify what you mean by non-classical Maass wave form. Commented Apr 15, 2013 at 8:34
• You're right that the term I used is shaky, but let's say anything non-holomorphic. For example, in the case of GSp4 we could refer to modular forms associated to generic representations gsp4.org/cat-browser.php?categoryID=199 Thanks for the response so far! Commented Apr 15, 2013 at 18:32

I don't think it is practical to directly find higher rank Maass forms along the lines of what Hejhal did, because they are functions of several variables and their Fourier expansions involve multiple sums. Not to mention the need to implement the appropriate special functions that appear. Even if you have the Fourier expansion on the computer (which has been done for SL(3,Z) (by Boris Mezhericher) but not Sp(4,Z)), using the relations to make equations in the coefficients and/or the analog of integrating along horocycles is probably computationally prohibitive.

• Thank you! Do you know of a reference for the SL(3, Z) case? Commented Apr 18, 2013 at 19:49
• I think the paper on his home page is the same as the one he posted to the ArXiv: sites.google.com/site/mboris Commented Apr 19, 2013 at 10:05