algorithm for reducing to the fundamental domain $\mathbb{H}/SL(2,\mathbb{Z})$ I have $y << 1$ and $x \in [0,1]$ uniformly chosen at random and I want to find its representative in the fundamental domain with $\big|\mathrm{Re} \; \tau \big|< \frac{1}{2}$ and $|\tau| > 1$.  

How do we find this representative? And how it related to the Euclidean algorithm.

Do you have an expression for the Möbius transformations that preserve this?
It appears that finding $m\colon\mathbb{H}\rightarrow\mathbb{H}$ such that
$0\mapsto 0$
$\frac{1+i\sqrt{3}}{2}\mapsto \frac{3+2i}{6}$
$\frac{1-i\sqrt{3}}{2}\mapsto \frac{1+i\sqrt{3}}{2}$
may help.
(I am estimating with 3+2i/6)
