  Given a word $w(x_{1},\ldots,x_{r})$, i.e., an element in the free group on
$r$ elements, and an integer $d\geq1$, we study the characteristic polynomial
of the random matrix $w(X_{1},\ldots,X_{r})$, where $X_{i}$ are Haar-random
independent $d\times d$ unitary matrices. If $c_{m}(X)$ denotes the $m$-th
coefficient of the characteristic polynomial of $X$, our main theorem implies
that there is a positive constant $\epsilon(w)$, depending only on $w$, such
that \[
\left|\mathbb{E}\left(c_{m}\left(w(X_{1},\ldots,X_{r})\right)\right)\right|\leq\left(\begin{array}{c}
d\\ m \end{array}\right)^{1-\epsilon(w)}, \] for every $d$ and every $1\leq
m\leq d$.
  Our main computational tool is the Weingarten Calculus, which allows us to
express integrals on unitary groups such as the expectation above, as certain
sums on symmetric groups. We exploit a hidden symmetry to find cancellations in
the sum expressing $\mathbb{E}\left(c_{m}(w)\right)$. These cancellations,
coming from averaging a Weingarten function over cosets, follow from Schur's
orthogonality relations.
