Expectation of trace of nth power of unitary matrices 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 $m>1$ ?
 A: This is an elaboration on my comment.
Let's start in a more general setting since the comment was not about the groups $U(n)$:
Let $G$ be a compact group and $H$ a normalized Haar measure on it.
Let $p_m:G\to G$ be defined by $p_m(x)=x^m$, $m\in\mathbb Z$.
Now for any continuous function $f:G\to\mathbb R$ we have
$$
\int_Gf(p_m(x))dH(x)
=
\int_Gf(x)d(p_m)_*H(x).
$$
Since $p_m$ is a homomorphism1, the pushforward $(p_m)_*H$ is a normalized Haar measure on $p_m(G)$.
If $p_m$ is surjective, it follows from the uniqueness of Haar measures that $p_{m*}H=H$ and thus
$$
\int_Gf(p_m(x))dH(x)
=
\int_Gf(x)dH(x).
$$
This idea made me think that the integral would be independent of $m$.

1
The map is not a homomorphism.
The argument works (I think) for homomorphisms, but typically $p_m$ is not a homomorphism in a nonabelian group.
If $p_m$ is not a homomorphism, the pushforward doesn't have to be a Haar measure and the proof falls apart.
Maybe I'll let this answer stay here as a warning example...
A: $$\int_{{\rm U}(n)} dU\,|{\rm Tr}\,(U^m)|^2={\rm min}\,(n,m).$$
see Theorem 2.1.b of Diaconis and Evans (2001). [*]
[*] This 2001 reference corrects an earlier paper by Diaconis and Shahshahani (1994), which would have given as an answer $m$ instead of ${\rm min}\,(n,m)$.
