Computation of weight 1 modular forms with exotic representations

TL;DR

This paper introduces an improved deterministic algorithm for computing weight 1 modular forms with exotic representations, enabling comprehensive analysis of such forms up to level 10,000.

Contribution

It presents a novel, optimized algorithm based on Hecke stability and trace formulas for computing weight 1 modular forms with exotic representations.

Findings

Computed all such forms with level ≤ 10,000

Determined dimensions of all weight 1 newform spaces up to level 10,000

Enhanced understanding of exotic representations in modular forms

Abstract

We present a deterministic algorithm for computing spaces of weight 1 modular forms with exotic representations. This algorithm is an improved version of Schaeffer's Hecke stability method, utilising the author's previous work on the twist-minimal trace formula for weight 2 holomorphic forms, and presenting a method of lifting forms from characteristic p. The algorithm was used to compute all such forms with level at most 10,000. Together with Sutherland's computation of dihedral forms, this allows us to present the dimensions of all weight 1 newform spaces up to level 10,000.