Vol. 2, 2019

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

Owen Colman, Alexandru Ghitza and Nathan C. Ryan

Vol. 2 (2019), No. 1, 207–220
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.

Keywords
Siegel modular forms, Hecke operators
Mathematical Subject Classification 2010
Primary: 11F46, 11F60
Milestones
Received: 2 March 2018
Revised: 17 June 2018
Accepted: 17 September 2018
Published: 13 February 2019
Authors
Owen Colman
School of Mathematics and Statistics
University of Melbourne
Parkville, VIC
Australia
Alexandru Ghitza
School of Mathematics and Statistics
University of Melbourne
Parkville, VIC
Australia
Nathan C. Ryan
Department of Mathematics
Bucknell University
Lewisburg, PA
United States