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.