Ramanujan type congruences for the Klingen-Eisenstein series

In the case of Siegel modular forms of degree $n$, we prove that, for almost all prime ideals $\mathfrak{p}$ in any ring of algebraic integers, mod ${\mathfrak{p}}^m$ cusp forms are congruent to true cusp forms of the same weight. As an application of this property, we give congruences for the Klingen-Eisenstein series and cusp forms, which can be regarded as a generalization of Ramanujan’s congruence. We will conclude by giving numerical examples.