Eigenspaces of Newforms with Nontrivial Character

Abstract

Let $S_k(\hat{I}{“}_0(N),\ddot{I}‡)$ denote the space of holomorphic cuspforms with Dirichlet character $\ddot{I}‡$ and modular subgroup $\hat{I}{“}_0(N)$. We will characterize the space of newforms $S_k^{new}(\hat{I}{“}_0(N),\ddot{I}‡)$ as the intersection of eigenspaces of a particular family of Hecke operators, generalizing the work of Baruch-Purkait to forms with non-trivial character. We achieve this by obtaining representation theoretic results in the $p$-adic case which we then de-adelizize into relations of classical Hecke operators.

18 pages