What does the quotient $\mathcal{H}/\Gamma_0(2)$ look like? We often see in books (typically of modular forms) the fundamental domain of the modular surface $\mathcal{H}/SL_2(\mathbb{Z})$. I would like to see similar pictures and know how to draw them. In particular, what does $\mathcal{H}/\Gamma_0(2)$ look like?

You might want to read about "Farey symbols". These are a compact way of describing fundamental domains for subgroups of $\operatorname{PSL}_2(\mathbb{Z})$ and how the edges are identified by the group generators. Sage or Magma can compute them for any congruence subgroup. There's a lovely survey article by Kurth and Long explaining the theory here: https://arxiv.org/abs/0710.1835.
