Maass cusp forms for large eigenvalues

TL;DR

This paper reports on the numerical computation of Maass cusp forms for the modular group with very large eigenvalues, providing new records in eigenvalue size and Fourier coefficients.

Contribution

The paper introduces a method to compute Maass cusp forms with eigenvalues exceeding 40,000, surpassing previous computational limits.

Findings

Fourier coefficients of two cusp forms with eigenvalues over 40,000

Largest eigenvalues computed for the modular group to date

Eigenvalues exceed 130 millionth eigenvalue

Abstract

We investigate the numerical computation of Maass cusp forms for the modular group corresponding to large eigenvalues. We present Fourier coefficients of two cusp forms whose eigenvalues exceed r=40000. These eigenvalues are the largest that have so far been found in the case of the modular group. They are larger than the 130millionth eigenvalue.