I will discuss the coefficients of the characteristic polynomial of a random unitary matrix. These are commonly known as secular coefficients, and their study was initiated by Diaconis and Gamburd in 2004 who showed that their moments are linked to the combinatorics of magic squares. I will discuss generalisations of this result and some new connections to random permutations and to Gaussian multiplicative chaos theory. This enables us to write down limit theorems for the coefficients in certain circumstances and to partially resolve an open problem of Diaconis and Gamburd who asked about this limit behaviour. This is joint work with Joseph Najnudel (Université Côte d'Azur) and Elliot Paquette (McGill University). Speaker(s): Nicholas Simm (University of Sussex)