Proportion of Atkin-Lehner sign patterns and Hecke Eigenvalue Equidistribution

Let $N\ge 1$, $k\ge 2$ even, and $\sigma$ denote a sign pattern for $N$. In this paper, we first determine the exact proportion of forms in $S_k(N)$ and $S_k^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)$ with a given Atkin-Lehner sign pattern $\sigma$. Then we study the asymptotic behavior of the Hecke operators $T_p$ over the subspaces of $S_k(N)$ and $S_k^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)$ with Atkin-Lehner sign pattern $\sigma$. In particular, for the $p$-adic Plancherel measure ${\mu }_p$, we show that the Hecke eigenvalues for $T_p$ over these subspaces are ${\mu }_p$-equidistributed as $N+k\to \mathrm{\infty }$.