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?



What is the rank of the period lattice of modular forms?
For Q1: I'm assuming your $f$ is a normalized Hecke *newform* of level $N$ with coefficients in $\mathbf{Z}$. Then the choice of $f$ determines a splitting of the homology as


$$H\_1(X\_0(N), \{cusps\}, \mathbf{Q}) = (f\text{-generalised eigenspace}) \oplus \text{(other stuff)}.$$


By the Eichler--Shimura isomorphism, we see that the $f$-generalized eigenspace is semisimple (i.e. it's a genuine eigenspace) and has dimension 2. When we integrate against $f$, it kills the "other stuff"; so the map $H\_1(X\_0(N), \{cusps\}, \mathbf{Z}) \to \mathbf{C}$ given by integration against $f$ factors through the image of $H\_1(X\_0(N), \{cusps\}, \mathbf{Z})$ in the $f$-eigenspace of $H\_1(X\_0(N), \{cusps\}, \mathbf{Q})$, and a little elementary linear algebra shows that this eigenspace quotient has to be free of rank 2.


So the image in $\mathbf{C}$ has rank *at most* 2. Since this lattice has non-trivial intersection with both $\mathbf{R}$ and $i \mathbf{R}$, its rank must be exactly 2.


For Q2, the comparison of lattices arising from "usual" and "relative" homology: one can show that the cokernel of the natural map from $H\_1(X\_0(N)) \to H\_1(X\_0(N), \{cusps\})$ is a free $\mathbf{Z}$-module of finite rank which is *Eisenstein* as a Hecke module, i.e. every Hecke eigenvalue system that shows up looks like an Eisenstein series (this is essentially the Manin--Drinfeld theorem). It follows that, for a given cuspidal eigenform $f$ as above, the only primes $p$ that can divide the index of the two period lattices are those such that $f$ is congruent mod $p$ to an Eisenstein series. If I remember correctly, you see this concretely with the unique weight 2 newform of level 11: it is congruent to an Eisenstein series mod 5, and the quotient between the two lattices is a cyclic group of order 5.


If you want to learn more, then you should read about *modular symbols*; the books by [John Cremona](https://johncremona.github.io/book/fulltext/index.html) and [William Stein](https://wstein.org/books/modform/modform/) are both excellent references.


