Let $\Gamma\subset SL(2,\mathbb{R})$ be a Fuchsian group of the first kind. Let $c\_1, c\_2$ be inequivalent cusps of $\Gamma.$ 
Consider $f\in M\_k(\Gamma)$ a weight $k$ holomorphic automorphic form, and suppose the Fourier expansion of $f$ at the cusp $c\_1$ is known. 


Given the above expansion, is there an algorithm to compute (even numerically) the Fourier expansion of $f$ at the cusp $c\_2$?
Thanks!



Fourier expansion at inequivalent cusps
This can be done numerically. The $n$-th Fourier coefficient around the cusp is given by an integral of the form $$\int\_0^{1} f|\_{\sigma\_2} (z) e(-nz) dx,$$ where $\sigma\_2$ is a scaling matrix for the cusp $c\_2$. See p.43 of Iwaniec's Topics in Classical Automorphic Forms for definitions. In the above, $z=x+iy$ and the integral is independent of $y>0$ which generally one picks to make the calculation as efficient as possible. Also, I am assuming that the multiplier system is such that $f|\_{\sigma\_2}$ is periodic with period $1$.


In principle, the Fourier expansion of $f$ allows one to numerically approximate $f|\_{\sigma\_2}$ at any $z \in \mathbb{H}$, so the above integral can be numerically calculated to any degree of precision.


added later: The same method works for non-congruence subgroups, as well as more general automorphic forms (e.g. Maass cusp forms).


