TITLE: U(3) Sato-Tate measure.
QUESTION [10 upvotes]: An undergraduate is performing some computations, related to a Sato-Tate conjecture of $U(3)$ type (a curve over $Q$, for which the roots of local L-functions look like eigenvalues of a random matrix in $U(3)$ and their complex conjugates, scaled appropriately).
There are well-known results in the literature, on random matrices from $U(3)$, with respect to Haar measure.  For example, the expected value of the trace is zero (obviously), and the expected value of the square of the trace is one (not-so-obviously, but this is what I recall).  The Weyl integration formula allows one to express other expectations as definite integrals.  Results of Diaconis and others give a few moments, but most results I find look at $U(N)$ as $N$ grows large.
Does anyone know a precise elementary formula for the distribution of traces of a random matrix in $U(3)$?  Or a good reference?
For example, the trace of a random matrix $g$ in $SU(2)$ is between $-2$ and $2$.  The probability distribution of $Tr(g) = t$ is given, up to some normalization, by:
$$P(t) = \sqrt{1 - \left( \frac{t}{2} \right)^2 },$$
The real part of the trace of a random matrix $g$ in $U(3)$ is between $-3$ and $3$.  Can anyone write the probability distribution of $Tr(g) = z$, in as simple a fashion?  Of course, here the traces can be complex -- I'd be happy also to find the probability that $Re(Tr(g)) = t$, as an elementary function of $t$.

REPLY [3 votes]: If the literature is such a poor state, I'm a little puzzled as to why. 
Maybe for SU(3) it should be clear in principle what is happening? The trace is an invariant of conjugacy classes, and the Weyl integration formula should reduce any issue to an integral over a two-dimensional region. Namely the maximal torus T is acted on by the Weyl group W, and apart from stuff of Haar measure zero the quotient can be represented by an explicit fundamental domain. The push forward of Haar measure on SU(3) has a density with respect to Lebesgue measure on the fundamental domain that was written down by Weyl (if not before). The trace is also a function on this fundamental domain. Points in T can be represented by three angles summing to zero mod 2pi. 
Am I being slow? The generic conjugacy class is not a difficult thing to understand, once having thrown out something of measure zero. This looks like an exercise in multiple integration over sets x + y + z = constant.