Hecke eigenvalues of Siegel modular forms of "different weights"

TL;DR

This paper investigates the Hecke eigenvalues of Siegel modular forms of different weights, providing explicit criteria for their eigenvalues and Fourier coefficients to be distinct, especially in degree two under certain conditions.

Contribution

It introduces explicit sets of Hecke eigenvalues and Fourier coefficients that must differ for Siegel eigenforms of different weights, advancing understanding of their spectral properties.

Findings

Explicit sets of Hecke eigenvalues are identified.

Conditions under which Fourier coefficients must differ are established.

Results are specialized to degree two forms with additional assumptions.

Abstract

Given two Siegel eigenforms of different weights, we determine explicit sets of Hecke eigenvalues for the two forms that must be distinct. In degree two, and under some additional conditions, we determine explicit sets of Fourier coefficients of the two forms that must be distinct.