Circles bounding $\mathrm{SL}(2,\mathbb{Z})$ fundamental domain translates

Question

The (closure of the) "canonical" fundamental domain for $\mathrm{SL}(2,\mathbb{Z})\backslash\mathbb{H}$ is given by $F=\{z=x+iy\in\mathbb{H}:2x\in[-1,1],\lvert z\rvert\ge1\}$. $\mathrm{SL}(2,\mathbb{Z})$ is generated by translation by $\mathbb{Z}$ along the real axis and inversion through the unit circle ($z\mapsto-\frac{1}{z}$). These are both conformal, so if we extend the bounding circles (including degenerate circles) we consider $\{x+iy:x=-\frac{1}{2}\}$, $\{x+iy:x=\frac{1}{2}\}$, $\{z\in\mathbb{H}:\lvert z\rvert=1\}$, and transforming these by $\mathrm{SL}(2,\mathbb{Z})$ sends these to other circles in $\mathbb{H}$ which meet the boundary $\mathbb{R}$ at two rational points. My question is: is there a known characterization of which circles arise from this group action on those three circles?