Let $X_0(N)$ be the usual modular curve. For $\chi$ a quadratic Dirichlet character of conductor $D$, define the homology class $c(\chi)=\sum_{i=0}^{|D|-1}\chi(i)\{\frac{i}{|D|}{\infty} \}$; here $\{a,b\}$ is the path joining the points $a,b$. When $D$ and $N$ are coprime, this construction yields a well-defined class in $H_1(X_0(N),\mathbf{Z})$ (see Vatsal's paper "Canonical periods and congruence formulae"). Reducing modulo a fixed prime $p$ gives a class in $H_1(X_0(N),\mathbf{F}_p$). My question is:

As $\chi$ varies over quadratic characters of discriminant prime to $N$, is it reasonable to expect that the classes $c(\chi)$ become uniformly distributed in $H_1(X_0(N),\mathbf{F}_p)$ with respect to counting measure?

My motivation for asking this is that pairing $c(\chi)$ with certain mod-$p$ cohomology classes essentially gives mod-$p$ special values of L-functions, so knowing this uniform distribution would imply a lot of things about vanishing and nonvanishing of L-functions modulo $p$.