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arXiv · 2608.18497

Asymptotics of Hecke polynomial coefficients on the Atkin-Lehner eigenspaces

Timothy Nelson·Erick Ross·Maya Wassercug·Hui Xue
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Abstract

Let $S_k^\sigma(N)$ denote the space of cusp forms of level $N$, weight $k$, and Atkin-Lehner sign pattern $\sigma$, and $S_k^{\mathrm{new},\sigma}$ denote its new subspace. In this paper, we study the asymptotic behavior of the coefficients of the $m$-th Hecke polynomial over $S_k^\sigma(N)$ and $S_k^{\mathrm{new},\sigma}(N)$. In particular, we show that in certain settings, all but finitely many of these coefficients take a particular sign. We also study settings in which the coefficients do not tend to any particular sign.

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Timothy Nelson, Erick Ross, Maya Wassercug, Hui Xue. 2026-08-19. Asymptotics of Hecke polynomial coefficients on the Atkin-Lehner eigenspaces. https://arxiv.org/abs/2608.18497

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