Newspaces with Nebentypus: An Explicit Dimension Formula and Classification of Trivial Newspaces

TL;DR

This paper provides an explicit dimension formula for newspaces of modular forms with Nebentypus, classifies cases with zero dimension, and explores the distribution of dimensions, disproving a related conjecture.

Contribution

It derives an explicit dimension formula for the newspace of modular forms with Nebentypus and classifies all cases with zero dimension.

Findings

Classified all triples where the newspace dimension is zero.

Proved that outside a specific infinite family, the dimension is bounded.

Disproved Greg Martin's conjecture on possible dimensions of the newspace.

Abstract

Consider $N \geq 1$, $k \geq 2$, and $\chi$ a Dirichlet character modulo $N$ such that $\chi(-1) = (-1)^k$. For any bound $B$, one can show that $\dim S_k(\Gamma_0(N),\chi) \le B$ for only finitely many triples $(N,k,\chi)$. It turns out that this property does not extend to the newspace; there exists an infinite family of triples $(N,k,\chi)$ for which $\dim S_k^{\text{new}}(\Gamma_0(N),\chi) = 0$. However, we classify this case entirely. We also show that excluding the infinite family for which $\dim S_k^{\text{new}}(\Gamma_0(N),\chi) = 0$, $\dim S_k^{\text{new}}(\Gamma_0(N),\chi) \leq B$ for only finitely many triples $(N,k,\chi)$. In order to show these results, we derive an explicit dimension formula for the newspace $S_k^{\text{new}}(\Gamma_0(N),\chi)$. We also use this explicit dimension formula to prove a character equidistribution property and disprove a conjecture from Greg Martin that $\dim S_2^{\text{new}}(\Gamma_0(N))$ takes on all possible non-negative integers.