 Solution

Index-$p$ overlattices ${\Lambda }^{\prime }$ correspond to order-$p$ subgroups of $(p^{-1}\Lambda )/\Lambda \cong {\mathbb{F}}_p^2$, hence to the $p+1$ lines in that vector space.

If $p\nmid N$, the order of $v+{\Lambda }^{\prime }$ cannot decrease: its decrease would have a factor dividing both $p=[{\Lambda }^{\prime }:\Lambda ]$ and $N$. Thus all $p+1$ overlattices are counted.

If $p\mid N$, the element $(N/p)v+\Lambda$ is a nonzero point of order $p$ in $\mathbb{C}/\Lambda$. Exactly one of the $p+1$ overlattices contains it; in that overlattice the image of $v$ has order $N/p$, while in every other one it retains order $N$. ThereforeThis is the Prime-index overlattices preserving a Gamma 1 level structure count.