Hecke Eigenvalue Equidistribution over the Newspaces with Nebentypus

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