Let $G=GL(2, F)$ and $K$ a maximal compact subgroup. Unramified Hecke operators are defined by the action of the double cosets $$T(n) = \bigcup_{\substack{ad=n, a>0 \\ a|d}} K \left( \begin{array}{cc} a & \\ & d \end{array} \right) K$$

There is a decomposition in right cosets $$\bigcup_{\substack{ad=n, a>0}} \bigcup_{b \mod d}K \left( \begin{array}{cc} a & b \\ & d \end{array} \right)$$

I would like to know the volumes of $T(n)$. So here are my questions:

is there a similar standard decomposition in left cosets?

can we directly compute the measure (for a fixed left Haar measure on $G$) of the right cosets?