This should be both well-known and probably easy, but I was wondering if the following is known (and, if so, how to easily calculate the thing or where to read about how to calculate it):

what is $$\int_{\mathrm{SU}(n)} \mathrm{tr}(U^k) dU?$$ (Here by "$dU$" I mean normalized Haar measure.)

Of course for $k$ not a multiple of $n$ the integral is zero. (It's always zero on $\mathrm{U}(n)$ by the same argument.) Also by Weyl's integration formula (i.e. averaging over conjugates of $U$) one immediately reduces to performing the integral over diagonal matrices, but I didn't see a way to get a clean answer out.

Apologies if this is easy! I just couldn't find a good reference.

Thanks!