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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 mth 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 for every d and every 
\[|\mathbb{E}(c_{m}(w(X_{1},\ldots,X_{r})))|\leq\binom{d}{m}^{\!\!1-\epsilon(w)},\]
$1\leq m\leq d$. 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 
$\mathbb{E}(c_{m}(w))$. These cancellations, coming from averaging a Weingarten function over cosets, follow from Schur’s orthogonality relations.