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There is a well known proof for finding the dimension of modular forms for the full modular group using the residue formula. It is on page 71 of these online notes:

http://www.math.ou.edu/~kmartin/mfs/ch5.pdf

I've read that this can be done for a general congruence group by finding a fundamental domain then performing the contour over the boundary in a similar way. I've been trying for days to do this for the congruence group $\Gamma_0(4)$ but am having no luck. The fundamental domain is given by this nice little Java app:

https://www.math.lsu.edu/~verrill/fundomain/index2.html

Any help would be much appreciated.

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You can prove the "valence formula", Theorem 5.14 of the notes you link, for a general congruence subgroup by explicit hands-on methods. But getting the dimension formula in general is more difficult. From the valence formula you have an upper bound, and for level SL2Z one can explicitly write down enough modular forms to show that it's an equality; but for general levels this won't work, partly because it's hard to prove the existence of lots of modular forms in a systematic way, and also because there is a correction term which the contour integral approach doesn't see: the genus of the Riemnn surface $X(\Gamma)$. So in the higher genus cases you need the machinery of the Riemann-Roch theorem.