Modular units and cuspidal divisor classes on $X_0(n^2M)$ with $n|24$ and $M$ squarefree

For a positive integer $N$, let $mathscr C(N)$ be the subgroup of $J_0(N)$ generated by the equivalence classes of cuspidal divisors of degree $0$ and $mathscr C(N)(mathbb Q):=mathscr C(N)cap J_0(N)(mathbb Q)$ be its $mathbb Q$-rational subgroup. Let also $mathscr C_{mathbb Q}(N)$ be the subgroup of $mathscr C(N)(mathbb Q)$ generated by $mathbb Q$-rational cuspidal divisors. We prove that when $N=n^2M$ for some integer $n$ dividing $24$ and some squarefree integer $M$, the two groups $mathscr C(N)(mathbb Q)$ and $mathscr C_{mathbb Q}(N)$ are equal. To achieve this, we show that all modular units on $X_0(N)$ on such $N$ are products of functions of the form $eta(mtau+k/h)$, $mh^2|N$ and $kinmathbb Z$ and determine the necessary and sufficient conditions for products of such functions to be modular units on $X_0(N)$.