  Let $g$ be a random matrix distributed according to uniform probability
measure on the finite general linear group $\mathrm{GL}_n(\mathbb{F}_q)$. We
show that $\mathrm{Tr}(g^k)$ equidistributes on $\mathbb{F}_q$ as $n \to
\infty$ as long as $\log k=o(n^2)$ and that this range is sharp. We also show
that nontrivial linear combinations of $\mathrm{Tr}(g^1),\ldots,
\mathrm{Tr}(g^k)$ equidistribute as long as $\log k =o(n)$ and this range is
sharp as well. Previously equidistribution of either a single trace or a linear
combination of traces was only known for $k \le c_q n$, where $c_q$ depends on
$q$, due to work of the first author and Rodgers.
  We reduce the problem to exhibiting cancellation in certain short character
sums in function fields. For the equidistribution of $\mathrm{Tr}(g^k)$ we end
up showing that certain explicit character sums modulo $T^{k+1}$ exhibit
cancellation when averaged over monic polynomials of degree $n$ in
$\mathbb{F}_q[T]$ as long as $\log k = o(n^2)$. This goes far beyond the
classical range $\log k =o(n)$ due to Montgomery and Vaughan. To study these
sums we build on the argument of Montgomery and Vaughan but exploit additional
symmetry present in the considered sums.
