Eigenspaces of Newforms with Nontrivial Character

TL;DR

This paper characterizes the eigenspaces of newforms with nontrivial Dirichlet characters in modular form spaces, extending previous work to include nontrivial characters through Hecke operator analysis.

Contribution

It generalizes the characterization of newforms as Hecke eigenspaces to cases with nontrivial characters, using representation theory and classical Hecke relations.

Findings

Characterization of newforms with nontrivial characters as Hecke eigenspaces.

Representation theoretic results in the p-adic setting.

De-adelization into classical Hecke operator relations.

Abstract

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.