We find the number $s_k(p,\Omega)$ of cuspidal automorphic representations of $\mathrm{ GSp}(4,\mathbb{ A}_{ \mathbb{ Q}})$ with trivial central character such that the archimedean component is a holomorphic discrete series representation of weight $k\ge 3$, and the non-archimedean component at $p$ is an Iwahori-spherical representation of type $\Omega$ and unramified otherwise. Using the automorphic Plancherel density theorem, we show how a limit version of our formula for $s_k(p,\Omega)$ generalizes to the vector-valued case and a finite number of ramified places.
Categories: math.NT