Valence formula for level 2 modular forms

Question

In this question the valence formula for modular functions on $\Gamma(2)$ $$\nu_{[1]}(f)+\nu_{[0]}(f) +\nu_{[\infty]}(f) + \sum_{[p] \in \mathbb{H}\backslash \Gamma_2} \nu_{[p]}(f) = \frac{k}{2} $$ is mentioned. Could someone please provide a proof sketch or a reference where this is proved, preferably using the argument principle as in the level 1 case (I am not familiar with the Riemann Roch theorem and don't need anything beyond the level 2 case). Which choice of fundamental domain do we typically use for level 2 modular forms? Thanks.