Maass forms and their $L$-functions

TL;DR

This paper explores Maass forms on Hecke congruence groups, providing eigenvalue examples and calculating associated L-functions, comparing results with random matrix theory predictions.

Contribution

It offers new explicit examples of Maass forms and their L-functions, including the first 1000 eigenvalues for a011, and compares these with theoretical predictions.

Findings

Low eigenvalues for Maass forms on a011 and small primes

First 1000 eigenvalues computed for a011

L-functions compared to random matrix theory predictions

Abstract

We present examples of Maass forms on Hecke congruence groups, giving low eigenvalues on $\Gamma_0(p)$ for small prime $p$, and the first 1000 eigenvalues for $\Gamma_0(11)$. We also present calculations of the $L$-functions associated to the Maass forms and make comparisons to the predictions from random matrix theory.