Proportion of Atkin-Lehner sign patterns and Hecke Eigenvalue Equidistribution

TL;DR

This paper determines the exact proportions of modular forms with specific Atkin-Lehner sign patterns and proves the equidistribution of Hecke eigenvalues over these subspaces as the level and weight grow large.

Contribution

It provides the first exact calculation of Atkin-Lehner sign pattern proportions and establishes Hecke eigenvalue equidistribution in this context.

Findings

Exact proportions of forms with given Atkin-Lehner sign patterns

Asymptotic Hecke eigenvalue equidistribution over these subspaces

Validation of conjectural distribution patterns for large levels and weights

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 behavior 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 $N+k \to \infty$.