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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?

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