I am trying to find the answer of 


$$\int dU \ |Tr(U^m)|^2$$


where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $\textit{U}(n)$ and $dU$ is a normalized Haar measure. In the case $m=1$, the answer seems to be $1$.


I don't know where to start. Does anyone have an idea? Is there a clean answer for $m>1$ ?



Expectation of trace of nth power of unitary matrices
$$\int\_{{\rm U}(n)} dU\,|{\rm Tr}\,(U^m)|^2={\rm min}\,(n,m).$$


see Theorem 2.1.b of [Diaconis and Evans](http://statweb.stanford.edu/~cgates/PERSI/papers/functionals.pdf) (2001). [\*]


[\*] This 2001 reference corrects an earlier paper by [Diaconis and Shahshahani](http://statweb.stanford.edu/~cgates/PERSI/papers/random_matrices.pdf) (1994), which would have given as an answer $m$ instead of ${\rm min}\,(n,m)$.


