Graphical cyclic supercharacters for composite moduli

Recent work has introduced the study of graphical properties of cyclic supercharacters, functions $\mathbb{Z}\mathrm{/}n\mathbb{Z}\to \mathbb{C}$ whose values are exponential sums with close connections to Gauss sums and Gaussian periods. Plots of these functions exhibit striking features, some of which have been previously explained when the modulus $n$ is a power of an odd prime. After reviewing this material, we initiate the graphical study of images of cyclic supercharacters in the case of composite $n$.