# Arithmetic behaviour of Hecke eigenvalues of Siegel cusp forms of degree two

## TL;DR

This paper investigates the simultaneous arithmetic properties of Hecke eigenvalues of Siegel cusp forms of degree two, focusing on forms that are not Saito-Kurokawa lifts and lie in distinct eigen spaces.

## Contribution

It provides new insights into the behavior of Hecke eigenvalues for non-lift Siegel cusp forms of degree two, expanding understanding beyond classical cases.

## Key findings

- Analysis of eigenvalue distributions for non-lift forms
- Distinct eigen space behavior of Hecke eigenvalues
- Implications for arithmetic properties of Siegel cusp forms

## Abstract

Let $F$ and $G$ be Siegel cusp forms for $\Sp_4(\Z)$ and weights $k_1, k_2$ respectively. Also let $F$ and $G$ be Hecke eigenforms lying in distinct eigen spaces. Further suppose that neither $F$ nor $G$ is a Saito-Kurokawa lift. In this article, we study simultaneous arithmetic behaviour of Hecke eigenvalues of these Hecke eigenforms.