  Diaconis and Gamburd computed moments of secular coefficients in the CUE
ensemble. We use the characteristic map to give a new combinatorial proof of
their result. We also extend their computation to moments of traces of
symmetric powers, where the same result holds but in a wider range.
  Our combinatorial proof is inspired by gcd matrices, as used by Vaughan and
Wooley and by Granville and Soundararajan. We use these CUE computations to
suggest a conjecture about moments of characters sums twisted by the Liouville
(or by the M\"obius) function, and establish a version of it in function
fields.
  The moral of our conjecture (and its verification in function fields) is that
the Steinhaus random multiplicative function is a good model for the Liouville
(or for the M\"obius) function twisted by a random Dirichlet character.
  We also evaluate moments of secular coefficients and traces of symmetric
powers, without any condition on the size of the matrix. As an application we
give a new formula for a matrix integral that was considered by Keating,
Rodgers, Roditty-Gershon and Rudnick in their study of the $k$-fold divisor
function.
