Why the field generated by the coefficients of a modular form over $\mathbb{Q}$ is finite over $\mathbb{Q}$

Question

I am going through modular forms and I came to know about that, the coefficients of a modular forms f over $\mathbb{Q}$ define a finite extension $K_f/Q$. Can anyone point me to the proof of this fact? I already know the coefficients are algebraic over $\mathbb{Q}$ from Milne's Notes, Proposition $5.27$. Is the same true for modular forms over $\mathbb{F}_p$?