Rigorous computation of Maass cusp forms of squarefree level

TL;DR

This paper presents a rigorous algorithm for computing and verifying Maass cusp forms of squarefree level, leveraging the Selberg trace formula, and applies it to generate extensive data supporting conjectures in the field.

Contribution

The paper introduces a new algorithm that rigorously computes Maass cusp forms of squarefree level using an explicit Selberg trace formula with Hecke operators.

Findings

Computed thousands of Maass forms across various levels

Provided numerical evidence supporting conjectures in the field

Validated the effectiveness of the new algorithm

Abstract

We derive an algorithm to rigorously compute and verify Maass cusp forms of squarefree level and trivial character. The main tool we use is an explicit version of the Selberg trace formula with Hecke operators due to Str\"{o}mbergsson. We use this algorithm to compute several thousand Maass forms for a range of levels and use this data to obtain numerical evidence towards various conjectures.