Generalisation of modular forms

Question

I am looking for a generalisation of a modular form that transforms as something like:

$f(\frac{a \tau+b}{c \tau+d}) = (c \tau+d)^k c^k f(\tau)$

I understand this cannot be literally true, as the multiplier c^k is not a root of unity, but does something like this arise in the context of modular forms? (or generalisations of those). Thanks!

Answer 1

The transformation should conform to the cocycle condition.

That is $f(\gamma_2\gamma_1\tau)=j(\gamma_2\gamma_1,\tau)f(\tau)$. But also, $f(\gamma_2\gamma_1\tau)=j(\gamma_2,\gamma_1\tau)f(\gamma_1\tau)=j(\gamma_2,\gamma_1\tau)j(\gamma_1,\tau)f(\tau)$. Thus, it must hold that $j(\gamma_2\gamma_1,\tau)= j(\gamma_2,\gamma_1\tau)j(\gamma_1,\tau)$. This holds for $j(\gamma,\tau)=(c\tau+d)^k$, but not for your sugesstion.

Answer 2

The only similar type formula I know follows from $$E_2(\frac{a\tau+b}{c\tau+d})-(c\tau+d)^2E_2(\tau)=-(6i/\pi)(c\tau+d)c\;,$$ where $E_2$ is the usual quasi-modular form of weight 2. You can raise that to the $k$th power if you want, but I don't see the point.

You also get similar formulas with derivatives of modular forms since the derivative of $(c\tau+d)^k$ is $k(c\tau+d)^{k-1}c$.