I recently learned about the set $$\mathrm{SL}_2(\Bbb Z)=\left\{\left({{a\quad b}\atop{c\quad d}}\right)\in\Bbb Z^{2\times 2}:ad-bc=1\right\}$$ where $\Bbb Z^{2\times 2}$ is the set of $2$ by $2$ integer matrices. I also learned of the set $$\Bbb H=\{z\in\Bbb C:\text{Im }z>0\}.$$ Then I learned of a set called the "orbit" of some $z\in\Bbb H$: $$\text{orb }z=\left\{\frac{az+b}{cz+d}:\left({{a\quad b}\atop{c\quad d}}\right)\in\mathrm{SL}_2(\Bbb Z)\right\}.$$ I have tried to visualize what $\text{orb }z$ looks like in the complex plane using Desmos, but Desmos does not really do complex numbers, and finding members of $\mathrm{SL}_2(\Bbb Z)$ is a little hard for me.
So, what do "orbits" look like? Does anyone have anything which I can use to visualize them? Thanks.