Textbooks talk at length of the modular properties of $\theta(z)$ or $\tau(z)$ and the prominent role of $SL(2,\mathbb{Z})$ or one of the congruence groups.

In that case, aren't the basic objects in elementary number theory such as the set of squares $\square = \{ n^2 : n \in \mathbb{Z}\}$ also related to the geometry of $\mathbb{H}/\Gamma_0(4)$ ?

Texts on modular forms say nothing about continued fractions, symbolic coding or the geodesic flow on the modular surface. Am I missing something here?