Invariant product with a weight-k real-analytic Eisenstein series
= Invariant product with a weight-k real-analytic Eisenstein series
If $f$ is a weight-$k$ modular form, then
$$
f(\tau)\overline{E_{k,s}(\tau)}(\operatorname{Im}\tau)^k
$$
is modular invariant. The factors transform by $j(\gamma,\tau)^k$, $\overline{j(\gamma,\tau)}^k$, and $|j(\gamma,\tau)|^{-2k}$.