I am trying to compute an integral which goes as:
$$I=\int_{\mathcal{F}}\left(\sqrt{Im(z)}\eta(z)\overline{\eta(z)}\right)^{-k}\frac{dzd\bar{z}}{Im(z)^2}$$
Where $\eta$ is Dedekind's $\eta$, $\mathcal{F}$ is the fundamental domain of the torus, and $k\in\mathbb{N}^*$.
I have been checking on procedures to solve it but wherever I go I end up crashing into 'If the function is an almost holomorphic modular form'. I have been reading and since it is modular invariant I only have to analyze the part $\frac{1}{|\eta(q)|^{2k}}$ (Where $q=e^{2\pi i z}$) to check that the integrand is indeed almost holomorphic; but if I understand everything correctly that coefficient is not an holomorphic function due to the absolut value, so I guess this integrand is not almost holomorphic. Am I right or am I mistaken somewhere?