On the Numerical Computation of Modular Forms

Abstract

This thesis presents a fast method for the numerical computation of modular forms. Our method significantly improves the performance of previous methods by using mixed-precision iterative solving techniques to reduce the number of iterations, and by optimizing the linear algebra using fast Fourier transforms and sophisticated polynomial evaluation algorithms. This makes it possible to compute a large number of modular forms on noncongruence subgroups. We also discuss how bases of modular forms and cusp forms can be computed from the Belyi map for genus zero subgroups, providing an efficient and rigorous method for these. In addition, we present a fast method for computing noncongruence Eisenstein series using Petersson products. We use this method to compute the first explicit examples of Eisenstein series on noncongruence subgroups and to study their algebraicity. We have created a database of modular forms, cusp forms, and Eisenstein series of weight <= 6 for noncongruence subgroups of index <= 17, which also contains the corresponding Belyi maps and elliptic curves. We also briefly discuss the numerical computation of other examples, such as Maass cusp forms and the generating function of traces of real singular moduli.