  We prove a harmonically weighted equidistribution result for the $p$-th
Satake parameters of the family of automorphic cuspidal representations of
$\operatorname{PGSp}(2n)$ of fixed weight $\mathtt{k}$ and prime-to-$p$ level
$N\to \infty$. The main tool is a new asymptotic Petersson formula for
$\operatorname{GSp}(2n)$ in the level aspect.
