A note on eigenvalues of Hecke operators on Siegel modular forms of degree two

Abstract

Let $F$ be a cuspidal Hecke eigenform of even weight $k$ on ${\operatorname {Sp} _4}(\mathbb {Z})$ with associated eigenvalues ${\lambda _m}(m \in \mathbb {N})$. Under the assumption that its first Fourier-Jacobi coefficient does not vanish it is proved that the abscissa of convergence of the Dirichlet series ${\sum _{m \geq 1}}\left | {{\lambda _m}} \right |{m^{ - s}}$ is less than or equal to $k$.