Computing the Satake p-parameters of Siegel modular forms

TL;DR

This paper introduces an algorithm to compute Satake p-parameters of Siegel modular forms from Hecke eigenvalues, providing explicit examples and exploring connections to the Ramanujan-Petersson Conjecture.

Contribution

The paper presents a novel algorithm for computing Satake p-parameters of Siegel modular forms from Hecke eigenvalues, advancing computational methods in automorphic forms.

Findings

Successfully computed Satake p-parameters for specific Siegel modular forms.

Established a link between computed parameters and the Ramanujan-Petersson Conjecture.

Provided explicit examples demonstrating the algorithm's effectiveness.

Abstract

Hecke eigenvalues of classical modular forms often encode a wealth of arithmetic data. The Satake $p$-parameters of a Siegel modular form play a role analogous to the one played by Hecke eigenvalues in the characterization of classical modular forms. In this paper we present an algorithm by which we can compute the Satake $p$-parameters of a Siegel modular form $F$ if we are given the Hecke eigenvalues of $F$ with respect to the generators of the local Hecke algebra. We compute explicit examples and relate our computations to the Ramanujan-Petersson Conjecture.