  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.
