  Let $S_{k}(\Gamma_0(N),\chi)$ denote the space of holomorphic cuspforms with
Dirichlet character $\chi$ and modular subgroup $\Gamma_0(N)$. We will
characterize the space of newforms $S_{k}^{new}(\Gamma_0(N),\chi)$ 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.
