Number of $SL_2(\mathbb{Z})$-translates lying within a fixed ray in $\mathfrak{h}$

Question

This is probably an easy question, but I can't seem get the right idea so here goes: Let $\tau \in \mathfrak{h}$ (where $\mathfrak{h}$ denotes the upper halfplane) be given. For any two positive real numbers $M_1 < M_2$, I am interested in calculating $$ |\{ \gamma \in SL_2(\mathbb{Z}) \ | \ M_1 < \frac{\Re(\gamma.\tau)}{\Im(\gamma.\tau)} < M_2 \ \}|$$ where $\gamma$ acts by fractional linear transformation. Any thoughts would be appreciated. Of course non-trivial upper/lower bounds would be of interest as well.