Effective bounds for Fourier coefficients of certain weakly holomorphic modular forms

In Jorgenson et al. (2016) [JST 16a], the authors derived generators for the function fields associated to certain low genus arithmetic surfaces realized through the action of the discrete Fuchsian group Γ0(N)+/{±1} on the upper half plane. In particular, they construct modular forms which are analogs to the modular discriminant and the Klein j-invariant of the full modular group PSL(2,Z). In this article, we produce effective and practical bounds for the Fourier coefficients in the q-expansion of such generators, thus allowing for rigorous numerical inspection of the generators.