of the oriented hyperbolic geodesic from $r$ to $s$, projected to $X_0(N)$.
paragraph
admin
Then $H_1(X_0(N),\operatorname{Cusps}_0(N);\mathbb{Z})$ is generated by the modular symbols $\{r,s\}$ with $r,s \in \mathbb{P}^1(\mathbb{Q})$. Moreover, if
paragraph
admin
\begin{align*}
S := \begin{pmatrix}0&-1\\ 1&0\end{pmatrix},
\qquad
U := \begin{pmatrix}0&-1\\ 1&-1\end{pmatrix},
\end{align*}