Eigenspaces of Newforms with Nontrivial Character

Let [Formula: see text] denote the space of holomorphic cuspforms with Dirichlet character [Formula: see text] and modular subgroup [Formula: see text]. We will characterize the space of newforms [Formula: see text] as the intersection of eigenspaces of a particular family of Hecke operators, generalizing previous work of Baruch–Purkait to forms with nontrivial character. We achieve this by obtaining representation theoretic results in the [Formula: see text]-adic case which we then de-adelize into relations of classical Hecke operators.