Hecke Eigenvalue Equidistribution over the Newspaces with Nebentypus

TL;DR

This paper studies the distribution of normalized Hecke eigenvalues over newspaces with nebentypus as the level and weight grow large, revealing their equidistribution properties.

Contribution

It establishes the equidistribution of normalized Hecke eigenvalues over newspaces with nebentypus as level and weight tend to infinity.

Findings

Eigenvalues become equidistributed in the limit

Normalized eigenvalues follow a specific distribution

Results extend previous unramified cases

Abstract

Fix a prime $p$, and let $\widehat T_p^{\mathrm{new}}(N,k,\chi) := \chi(p)^{-1/2} p^{-(k-1)/2} T_p^{\mathrm{new}}(N,k,\chi)$ denote the normalized $p$'th Hecke operator over the newspace with nenbentypus $S_k^{\mathrm{new}}(N,\chi)$. In this paper, we determine the distribution of the eigenvalues of $\widehat T_p^{\mathrm{new}}(N,k,\chi)$ as $N+k \to \infty$.