On the distribution of Satake parameters for Siegel modular forms

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