  For a positive integer $m$ and a subgroup $\Lambda$ of the unit group
$(\mathbb{Z}/m\mathbb{Z})^\times$, the corresponding generalized Kloosterman
sum is the function $K(a,b,m,\Lambda) = \sum_{u \in \Lambda}e(\frac{au +
bu^{-1}}{m})$. 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.
