Cusp widths (awaiting review)

For an open subgroup $H$ of $\GL_2(\widehat \Z)$ with $\det(H) =\widehat \Z^\times$, the cusps of $H$ are the double cosets $\pm H \backslash \GL_2(\widehat \Z) /\langle \delta\rangle$, where $ \delta := \begin{bmatrix}1&1\\0&1\end{bmatrix}$. These correspond to the cusps of the modular curve $X_H$.

The width of a cusp $c \in \pm H \backslash \GL_2(\widehat \Z) /\langle \delta\rangle$ is the number of right cosets it contains, or equivalently the smallest integer $h$ such that $c\delta^h = c$. The sum of the cusp widths is equal to the $\PSL_2$-index of $H$.