# group theory – Coset Representatives for \$U(2,mathbb{Z})\$ in \$SL(2,mathbb{Z})\$

If $$SL(2,mathbb{Z})$$ denotes the modular group and if $$U(2,mathbb{Z})$$ denotes the subgroup of upper triangular matrices with diagonal entries equal to $$1$$, then is there a (nice) description of distinct coset representatives for $$U(2,mathbb{Z})$$ inside $$SL(2,mathbb{Z})$$?

I tried using the free product decomposition of $$SL(2,mathbb{Z})$$ in terms of $$S$$ and $$ST$$ (where $$S$$ is the “inversion” and $$T$$ is the “translation”), but that is not leading me anywhere. Any help would be appreciated.