Hecke Eigenvalue Equidistribution over the Newspaces with Nebentypus

Abstract

Fix a prime $p$, and let ${\hat{T}}_p^{\mathrm{new}}(N,k,\ddot{I}‡):=\ddot{I}‡(p{)}^{-1/2}{p}^{-(k-1)/2}{T}_p^{\mathrm{new}}(N,k,\ddot{I}‡)$ denote the normalized $p$'th Hecke operator over the newspace with nenbentypus $S_k^{\mathrm{new}}(N,\ddot{I}‡)$. In this paper, we determine the distribution of the eigenvalues of ${\hat{T}}_p^{\mathrm{new}}(N,k,\ddot{I}‡)$ as $N+k\to \infty$.