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!



Generalisation of modular forms
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.


