Cusp widths (awaiting review)

For an open subgroup $H$ of $\GL_2(\widehat \Z)$, the cusps of $H$ are the double cosets $\SL_2(\widehat \Z)\backslash \bigl(\pm H \cap \SL_2(\widehat \Z)\bigr)/\langle \delta\rangle$, 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 is the number of right cosets it contains. The sum of the cusp widths is equal to the $\PSL_2$-index of $H$.