Dimensions of spaces of modular forms

TL;DR

This paper proves a conjecture about the distribution of dimensions of spaces of modular forms across various weights and levels, providing new insights into their value distribution.

Contribution

It establishes the conjecture of Ross and extends the results to general weights and different types of modular form spaces.

Findings

Confirmed Ross's conjecture on dimension distribution

Extended results to all even weights and various modular form spaces

Provided new understanding of the value distribution of modular form dimensions

Abstract

We prove a conjecture of Ross concerning the value distribution of $\dim S_2^{\rm new}(\Gamma_0(N))$ for $N\in\mathbb{N}$, as well as analogous results for general weight $k\in2\mathbb{N}$ and the full and twist-minimal spaces $S_k(\Gamma_0(N))$, $S_k^{\rm min}(\Gamma_0(N))$.