Computational aspects of Maass cusp forms

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Modular forms are certain holomorphic complex-valued functions with many symmetry properties. Some notable applications include monstrous moonshine and string theory (work that earned Borcherds the fields medal in 1998), Wiles' solution of Fermat's last theorem, and recent novel applications to sphere packing by Viazovska et al. Maass forms are closely related to modular forms, however certain conditions like holomorphy are no longer required which then makes them harder to compute. A notable application of Maass forms is in the solution to the question of what integers can be represented as the sum of three squares.Hejhal introduced an algorithm to find Maass cusp forms in 1990's, however this relies on a heuristic argument and has not been proven rigourously to converge in general for congruence subgroups. Some advances from a decade ago demonstrate that rigorous computation of Maass forms is possible from a different method using the Selberg trace formula, however this has only been carried out in the simplest case of the modular group.An effort to compute a large, rigorous, database of holomorphic modular forms for the L-functions and modular forms database (LMFDB) was completed in 2016, and already has several citations. The main aim for this project is provide a method to compute and certify a large database of Maass forms for arbitrary level and character. The project will explore some novel computation and certification methods based on the Selberg trace formula, namely with relation to Hecke operators. All relevant data will be published in the LMFDB.