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arXiv · 2511.13969

Proportion of Atkin-Lehner sign patterns and Hecke Eigenvalue Equidistribution

Erick Ross·Alexandre van Lidth·Martha Rose Wolf·Hui Xue
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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 $N+k \to \infty$.

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Erick Ross, Alexandre van Lidth, Martha Rose Wolf, Hui Xue. 2025-11-17. Proportion of Atkin-Lehner sign patterns and Hecke Eigenvalue Equidistribution. https://arxiv.org/abs/2511.13969

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