  Using the relationship between Siegel cusp forms of degree $2$ and cuspidal
automorphic representations of $\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}})$, 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 $2$.
