 Elliptic stabilizers of the modular group

For the boundary convention $-1/2<\mathrm{Re}z\le 1/2$, retaining only the right half of the unit circle, the exceptional points are $i$ and ${\rho }_{+}=e^{i\pi /3}$. Their stabilizers in $S{L}_2(\mathbb{Z})$ are the cyclic groups generated by $\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ and $\left(\begin{array}{cc}1& -1\\ 1& 0\end{array}\right)$, of orders four and six. Every other stabilizer is $\{I,-I\}$.