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: $$P(t) = 1 - t^2/4,$$

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 $t$.

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 $t$.