It is well-known that the $j$-invariant is a Hauptmodul for $SL(2,\mathbb Z)$ and has nonnegative integer Fourier coefficients. A reason for this is that the coefficients can be written as non-negative integer linear combinations of the dimensions of irreducible representations of the Fischer–Griess monster $\mathbb M$ group: The famous monstrous moonshine. It was later observed that characters of $\mathbb M$ are replicable modular functions for certain Fuchsian groups. However these modular functions do not all possess the property that their Fourier-coefficients are nonnegative.
I was wondering to what extent it is known for which genus zero groups there exist Hauptmoduln with purely non-negative integral Fourier-coefficients. Any recommendation on the literature would be most helpful!