Multiplicative relations for Fourier coefficients of degree 2 Siegel eigenforms

TL;DR

This paper establishes multiplicative relations among Fourier coefficients of degree 2 Siegel eigenforms and derives formulas for their eigenvalues, linking Fourier coefficients and eigenvalues in new ways.

Contribution

It introduces novel multiplicative relations for Fourier coefficients of degree 2 Siegel eigenforms and provides explicit formulas for their eigenvalues based on Fourier data and weight.

Findings

Proved multiplicative relations for Fourier coefficients.

Derived formulas for eigenvalues in terms of Fourier coefficients.

Expressed eigenvalues for Eisenstein series explicitly.

Abstract

We prove multiplicative relations between certain Fourier coefficients of degree 2 Siegel eigenforms. These relations are analogous to those for elliptic eigenforms. We also provide two sets of formulas for the eigenvalues of degree 2 Siegel eigenforms. The first evaluates the eigenvalues in terms of the form's Fourier coefficients, in the case $a(I) \neq 0$. The second expresses the eigenvalues of index $p$ and $p^2$, for $p$ prime, solely in terms of $p$ and $k$, the weight of the form, in the case $a(0)\neq 0$. From this latter case, we give simple expressions for the eigenvalues associated to degree 2 Siegel Eisenstein series.