On the modular integrals and their Mellin transforms

It is shown that if ${f}$ is an entire modular integral on $\Gamma(1)$ of weight k, with multiplier system $\nu$, then ${f}^m(\tau^r)$ is an entire modular integral on $\Gamma(1)$ of weight mk, with multiplier system $\nu$. A more general formula is obtained for the Mellin transforms. Some relations among the Mellin transforms of functions ${f}(\tau)$, ${f}^m(\tau^r)$ and ${f}^m(\tau^r/m^,,\chi)$ are also deduced.