Fourier expansions at cusps

Abstract

In this article we study the fields generated by the Fourier coefficients of modular forms at arbitrary cusps. We prove that these fields are contained in certain cyclotomic extensions of the field generated by the Fourier coefficients at infinity. We also show that this bound is tight in the case of newforms with trivial Nebentypus. The main tool is a result of Shimura on the interplay between the actions of $\mathrm{GL}_2^+(\mathbb{Q})$ and $\mathrm{Aut}(\mathbb{C})$ on modular forms.