Let $f$ be a weight $2$ cusp form for the group $\Gamma_0(N)$. I was experimenting with integrals of the form $$ \int_r^s f(z) \, dz$$ where $r, s \in \mathbf{P}^1(\mathbf{Q})$ and the integral above is over the geodesic in the upper-half plane connecting $r$ and $s$. When I plotted these period integrals, I noticed that the resulting values formed a rank $2$ lattice in $\mathbf{C}$. I have two questions about this:

1. Why would the above period lattice have rank $2$? Since we are integrating $f(z) \, dz$ over paths in the relative homology group $H^1(X_0(N), \{\text{cusps}\}, \mathbf{Z})$, I would expect the rank of the period lattice to grow with the rank of this homology group, because we integrating over more loops. Why is the rank of the period lattice always $2$, even if the rank of the homology group is bigger?

2. Inside the period lattice obtained from integrating over $H^1(X_0(N), \{\text{cusps}\}, \mathbf{Z})$, we can consider the sublattice from integrating over just the ordinary homology group $H^1(X_0(N), \mathbf{Z})$, omitting paths that are not closed loops in $X_0(N)$. (Equivalently, the sublattice is formed by only plotting $\int_r^s f(z) \, dz$ when $r$ and $s$ are $\Gamma_0(N)$-equivalent.) What is the index of this sublattice inside the full period lattice? Can one answer this in general?

Let $f$ be a weight $2$ cusp form for the group $\Gamma_0(N)$. I was experimenting with integrals of the form $$ \int_r^s f(z) \, dz$$ where $r, s \in \mathbf{P}^1(\mathbf{Q})$ and the integral above is over the geodesic in the upper-half plane connecting $r$ and $s$. When I plotted these period integrals, I noticed that the resulting values formed a rank $2$ lattice in $\mathbf{C}$. I have two questions about this:

1. Why would the above period lattice have rank $2$? Since we are integrating $f(z) \, dz$ over paths in the relative homology group $H^1(X_0(N), \{\text{cusps}\}, \mathbf{Z})$, I would expect the rank of the period lattice to grow with the rank of this homology group. Why is the rank of the period lattice always $2$, even if the rank of the homology group is bigger?

2. Inside the period lattice obtained from integrating over $H^1(X_0(N), \{\text{cusps}\}, \mathbf{Z})$, we can consider the sublattice from integrating over just the ordinary homology group $H^1(X_0(N), \mathbf{Z})$, omitting paths that are not closed loops in $X_0(N)$. (Equivalently, the sublattice is formed by only plotting $\int_r^s f(z) \, dz$ when $r$ and $s$ are $\Gamma_0(N)$-equivalent.) What is the index of this sublattice inside the full period lattice? Can one answer this in general?