Congruences for dimensions of spaces of Siegel cusp forms and $4$-core partitions

Using the relationship between Siegel cusp forms of degree $2$ and cuspidal automorphic representations of $\mathrm{G}\mathrm{S}\mathrm{p}(4,{\mathbb{A}}_{\mathbb{Q}})$, we derive some congruences involving dimensions of spaces of Siegel cusp forms of degree $2$ and the class number of $\mathbb{Q}(\sqrt{-p})$. We also obtain some congruences between the $4$-core partition function $c_4(n)$ and dimensions of spaces of Siegel cusp forms of degree $2$.