Note on Fourier expansions at cusps

Abstract

This was originally an appendix to our paper `Fourier expansions at cusps' [arXiv:1807.00391]. The purpose of this note is to give a proof of a theorem of Shimura on the action of $\mathrm{Aut}(\mathbb{C})$ on modular forms for $\Gamma(N)$ from the perspective of algebraic modular forms. As the theorem is well-known, we do not intend to publish this note but want to keep it available as a preprint.