The following question was suggested by Sasha's answer to the following question : Is the derivative of a modular function a modular function .
Question. What are the modular functions with respect to $\Gamma(n)$ of weight zero?
I know that the $j$-invariant is of weight zero with respect to $\Gamma(1)$.
Is the modular function $\lambda$ of weight zero with respect to $\Gamma(2)$?
More generally, is a Hauptmodul for $\Gamma(n)$ of weight zero?