The characteristic polynomial of a random unitary matrix is a much studied and mathematically rich object in random matrix theory. In this talk I will discuss the secular coefficients, those obtained after expanding the characteristic polynomial in terms of its monomial powers. These coefficients turn out to have interesting structure, related to combinatorial objects known as magic squares and to a holomorphic counterpart of Gaussian multiplicative chaos. I will discuss recent work where we obtain their limiting distributions in the Circular \beta Ensemble, for any \beta > 2. This is joint work with Joseph Najnudel, Elliot Paquette and Truong Vu.