  Let (G, V) be a prehomogeneous vector space, let O be any G(F_q)-invariant
subset of V(F_q), and let f be the characteristic function of O. In this paper
we develop a method for explicitly and efficiently evaluating the Fourier
transform of f, based on combinatorics and linear algebra. We then carry out
these computations in full for each of five prehomogeneous vector spaces,
including the 12-dimensional space of pairs of ternary quadratic forms. Our
computations reveal that these Fourier transforms enjoy a great deal of
structure, and sometimes exhibit more than square root cancellation on average.
  These Fourier transforms naturally arise in analytic number theory, where
explicit formulas (or upper bounds) lead to sieve level of distribution results
for related arithmetic sequences. We describe some examples, and in a companion
paper we develop a new method to do so, designed to exploit the particular
structure of these Fourier transforms.
