Effective Hecke eigenvalue equidistribution over the Atkin--Lehner subspaces

For a fixed prime $p$, let $μ_p$ denote the $p$-adic Plancherel measure. Then the first main goal of this paper is to prove effective (and moreover explicit) $μ_p$-equidistribution of the $p$-th Hecke eigenvalues over the Atkin--Lehner subspaces $S_k^σ(N) \subseteq S_k(N)$ and $S_k^{\operatorname{ne