I will discuss the arithmetic significance of Fourier
expansions of modular forms at cusps. After a brief introduction of new
methods of calculating these expansions, I will talk about joint work
with F. Brunault, where we determine the number field generated by
Fourier coefficients of newforms at a cusp. Finally I will show work in
progress on bounding the denominators of Fourier expansions at cusps and
applications of these bounds to a conjecture on the Manin constants of
elliptic curves.