Let $\Gamma$ be a normal subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$.
Let $\mathbb{H}$ be the upper half-plane.
Let $C$ be the set of all cusps of $\Gamma$.
Let $R = (\mathbb{H}\cup C)/\Gamma$ be the associated compact Riemann surface.
It is well-known that there is a correspondence between the modular forms of dimension $-2$ for $\Gamma$ and differentials on $R$. (See, Elliptic Modular Functions, An Introduction, B. Schoenberg, Chapter V, Pages 125 and 126).
The real questions are, how to prove that:
- Differentials on $R$ having no poles outside the cups and degree at most $-1$ at the cusps are represented by entire modular forms of dimension $-2$?
- A differential has a residue at a cusp $\alpha$ if the corresponding form does not vanish at $\alpha$?
One can take a look at Asymptotic Winding of the Geodesic Flow on Modular Surfaces and Continuous Fractions. Y. Guivarc'h and Y. Le Jan. Page 26.