  A linear equation with coefficients in $\mathbb{F}_q$ is common if the number
of monochromatic solutions in any two-coloring of $\mathbb{F}_q^n$ is
asymptotically (as $n \to \infty$) at least the number expected in a random
two-coloring. The linear equation is Sidorenko if the number of solutions in
any dense subset of $\mathbb{F}_q^n$ is asymptotically at least the number
expected in a random set of the same density.
  In this paper, we characterize those linear equations which are common, and
those which are Sidorenko. The main novelty is a construction based on choosing
random Fourier coefficients that shows that certain linear equations do not
have these properties. This solves problems posed in a paper of Saad and Wolf.
