A database of rigorous Maass forms

TL;DR

This work presents a database of rigorously computed Maass cuspforms on congruence subgroups $Γ_0(N)$. It combines three complementary algorithms—the Rigorous Trace Formula, Quasimode construction, and Rigorous Hejhal method—to certify eigenvalues $λ$ and Fourier coefficients $a_j(n)$ with provable bounds. The authors provide a large-scale dataset (35,416 forms) for squarefree levels up to $105$, including eigenvalues, the first 1000 coefficients, and visual portraits, stored with explicit error tolerances and integrated into the LMFDB. The approach prioritizes rigorous validation of spectral data, acknowledges current limitations (squarefree level, absence of $L$-functions), and outlines plans to generalize to broader levels and include $L$-functions in future work. This database thus serves as a valuable, verifiable resource for researchers in automorphic forms and related computational number theory.

Abstract

We announce a database of rigorously computed Maass forms on congruence subgroups $Γ_0(N)$ and briefly describe the methods of computation.