Proportion of Atkin–Lehner sign patterns and Hecke eigenvalue equidistribution

Let N ≥ 1 , k ≥ 2 even, and σ denote a sign pattern for N . In this paper, we first determine the exact proportion of forms in Sₖ(N) and Sₖⁿᵉʷ(N) with a given Atkin–Lehner sign pattern σ . Then we study the asymptotic behaviour of the Hecke operators Tₚ over the subspaces of Sₖ(N) and Sₖ^new(N) with Atkin–Lehner sign pattern σ . In particular, for the p -adic Plancherel measure μₚ , we show that the Hecke eigenvalues for Tₚ over these subspaces are μₚ -equidistributed as N+k → ∞ .