Visual properties of generalized Kloosterman sums

For a positive integer m and a subgroup Λ of the unit group (Z/mZ)×, the corresponding generalized Kloosterman sum is the function K(a,b,m,Λ)=∑u∈Λe(au+bu−1m) for a,b∈Z/mZ. Unlike classical Kloosterman sums, which are real valued, generalized Kloosterman sums display a surprising array of visual features when their values are plotted in the complex plane. In a variety of instances, we identify the precise number-theoretic conditions that give rise to particular phenomena.