The main result of this note is a characterization of those generalized modular functions of weight zero on $\Gamma_0(N)$ that have empty divisor, in terms of the growth of the exponents in their $q$-product expansion.
"On certain generalized modular forms." Funct. Approx. Comment. Math. 43 (1) 23 - 29, September 2010. https://doi.org/10.7169/facm/1285679143