Ideal orbits of toral automorphisms are the simplest (and arguably the most interesting) among the periodic orbits of fully chaotic maps. We show that for an ideal orbit the coordinate and momentum evolve independently, and that their dynamical evolution is determined by multiplication by a fixed integer modulo a prime. Their spectrum is found to be the product of a regular and an irregular function. The former is computed explicitly, while the latter is a sum of Dirichlet characters, whose irregular nature is the source of spectral fluctuations. We evaluate some statistical properties of these sums, and estimate the average spectral density of the phase coordinates. We show how to select toral automorphisms and initial conditions corresponding to ideal orbits of arbitrarily large period having certain prescribed properties. These results may be relevant to the problem of generating pseudorandom sequences.