Fourier coefficients of a modular form of higher level at a cusp other than $i\infty$ I've been trying to learn a bit about modular forms, and mainly using the Sherman-Diamond textbook. Now, looking at modular forms of higher level where we might have more than one cusp of the compactified modular curve, I was wondering how I might get an expansion of a given modular form around a given cusp, similarly to how we get them around $\tau = i\infty$, or equivalently $q = e^{2\pi i \tau} = 0$?
Specifically, are the expansions around different cusps related to the one around infinity? I was thinking that there must be a way to permute cusps and so to relate these expansions.
 A: For $f \in M_{2k}(\Gamma_0(N))$ you can say that $\displaystyle\frac{f(z)}{j'(z)^k}$ is meromorphic on $X_0(N)$ so that $$\frac{f(z)}{j'(z)^k}=h(j(z),j(Nz))\qquad \text{with }\quad h\in \Bbb{C}(x,y)$$
Once you know the coefficients of this rational function $h$ then for any $\alpha\in SL_2(\Bbb{Z})$, $$(c_\alpha z+d_\alpha)^{-2k} f(\alpha z) =  h(j(z),j(N \alpha z)) (c_\alpha z+d_\alpha)^{-2k}j'(\alpha z)^k$$ $$= h(j(z),j(\frac{Az+B}{D})) j'(z)^k$$

Where the Fourier expansion at $i\infty$ of the RHS is found from that of $j$ (which is known from its expression in term of $E_4,E_6$).

If $f\in M_{2k}(\Gamma(N))$ then it is the same idea except that $\displaystyle\frac{f(z)}{j'(z)^k}$ will be a rational function in $j$ and all the $j(\frac{az+b}{d})$, $ad=N,b\in 0\ldots d-1$.
If $f$ has an odd weight then you can do the same for $f^2$ and deduce from it the Fourier expansion of $f$ at each cusp.
So morally the problem gets relatively easy once you algebraized the (even weight) modular forms as some $k$-tensor of $1$-forms with no pole.

