Connections between Modular forms and Hyperbolic geometry

Question

I am taking a course each on Modular forms and Hyperbolic geometry currently and I have begun to like the nice connections that exists between them. I am still a beginner in both these subjects and would like to learn more. Here, I record a few of my observations regarding the connections. It would be of immense help if you can comment on these and give some helpful suggestions.

1. First and foremost is the plane of action and the group acting on it. In both the cases, we consider the group of isometries on $\mathbb{H}$ or its subgroup (the modular group $\operatorname{SL}_2(\mathbb{Z})$, with the plane of action being $\mathbb{H}$.
2. The action being properly discontinuous (Prop 2.1.1 in Diamond and Shurman) also holds a special place as in the case Hyperbolic geometry, it gives a characterization of Fuchsian groups and in Modular forms it is used in observing some topological properties of the modular curve Y(1).
3. I find the fixed points of a matrix and matrices which fix a particular point (basically stabilizer) to be an interesting connection as well as it helps one to classify the matrices into translations, dilation and rotations - based on the number of fixed points and where they lie. In the case of modular forms, classifying the points based on their stabilizers as elliptic and non elliptic helps in defining coordinate charts to prove that the modular curve is a Riemann surface.
4. The orbit of a Fuchsian group and its limit points are found in the boundary of $\mathbb{H}$. Similarly, the role of limit points in the case of modular forms is played by the cusps, adjoining which acts as a compactification of the modular curve and also helps in finding the dimension of the space of modular forms of a particular weight.
5. The Fundamental domain of $\operatorname{SL}_2(\mathbb{Z})$ that is the region {$z \in \mathbb{H} $ :|$\Re(z)$|<$\frac{1}{2}$ and |z|$\geq 1$} is an hyperbolic triangle. I searched about the fundamental domains of a few other subgroups too and it looks like they are also hyperbolic polygons. Is it true in general that, the fundamental domain of any congruence subgroup of $\operatorname{SL}_2(\mathbb{Z})$ is a hyperbolic polygon?

I would like to know more on the interplay between Modular forms and Hyperbolic geometry. So, it would be beneficial for me if you can give some more of such connections or direct me to a resource where I can see more of it.