Determinant-n matrix representatives for Hecke operators (${\Pi }_n$)

Left $S{L}_2(\mathbb{Z})$-orbits of integral matrices of positive determinant $n$ have unique representatives $\left(\begin{array}{cc}a& b\\ 0& d\end{array}\right)$ with $a,d>0$, $ad=n$ and $0\le b<d$. Bezout's identity reduces the first column to $(a,0)$; a row shear reduces $b$ modulo $d$.