Proportion of Atkin–Lehner sign patterns and Hecke eigenvalue equidistribution

Abstract 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 $S_k^{\mathrm{new}}(N)$ with a given Atkin–Lehner sign pattern $\sigma$ . Then we study the asymptotic behaviour of the Hecke operators $T_p$ over the subspaces of $S_k(N)$ and $S_k^{{\mathrm{new}}}(N)$ 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 $N+k \to \infty$ .