  In [6], Kohnen proves that if $\Gamma=\Gamma_0(N)$ where $N$ is a square-free
integer, then any modular function of weight $0$ for $\Gamma$ having a divisor
supported at the cusps is an $\eta$-product. Under the condition of having
rational Fourier coefficients, we are able to extend Kohnen's result to the
case where $N$ is the square of a prime. If the rationality condition does not
hold, we show that the statement is no longer true by providing a family of
counter-examples that generalizes naturally the Dedekind $\eta$-function. This
paper fits within the framework of generalized modular forms in the sense of
Knopp and Mason.
