Analytic evaluation of Hecke eigenvalues for Siegel modular forms of degree two

TL;DR

This paper introduces a new numerical method for evaluating Hecke eigenvalues of Siegel modular forms of degree two, which is more efficient than traditional Fourier coefficient-based approaches and allows adjustable precision.

Contribution

The paper presents a novel numerical evaluation technique for Hecke eigenvalues that improves efficiency and flexibility over standard Fourier coefficient methods.

Findings

The new method is more efficient than the standard approach.

It allows adjustable precision for eigenvalue approximations.

Potential for further optimization by selecting evaluation points.

Abstract

The standard approach to evaluate Hecke eigenvalues of a Siegel modular eigenform F is to determine a large number of Fourier coefficients of F and then compute the Hecke action on those coefficients. We present a new method based on the numerical evaluation of F at explicit points in the upper-half space and of its image under the Hecke operators. The approach is more efficient than the standard method and has the potential for further optimization by identifying good candidates for the points of evaluation, or finding ways of lowering the truncation bound. A limitation of the algorithm is that it returns floating point numbers for the eigenvalues; however, the working precision can be adjusted at will to yield as close an approximation as needed.