Suppose I have a compact Riemann surface of $g>1$ given by the quotient $H/\Gamma$ where I do know $\Gamma$ explicit. Is there a way to write down the power series of meromorphic functions, differentials, quadratic differentials, and so on explicitly if one does know these sections explicitly (for example in a hyperelliptic picture of the surface). I know that there is the subject of automorphic forms, but the literature I have seen about this concentrates on modular groups. Moreover it is typically written for number theorists. Is there literature for compact surfaces (for example $Y^2=Z^6-1$), and maybe readable for geometers.

Thank you.