cv.complex variables - Power series for meromorphic differentials on compact Riemann surfaces

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.