For a positive integer $m$ how does one "easily" calculate the integral $$\int_{\mathrm{SU}(n)}\mathrm{Tr}(g)^mdg?$$ Here $dg$ is the Probability Haar measure of $\mathrm{SU}(n)$. By "easily", I mean without doing messy calculation, for e.g. using Weyl integration formula (which seems to be very cumbersome). Checking the case $n=2$ it seems the answer should be $\delta_{n\mid m}$.
Thanks for any help!