  We prove that for any {\it fixed} trigonometric polynomial potential
satisfying a genericity condition, the spectrum of the two dimension periodic
Schr\"odinger operator has finite multiplicity and the Fourier series of the
eigenfunctions are uniformly exponentially localized about a finite number of
frequencies. As a corollary, the $L^p$ norms of the eigenfunctions are bounded
for all $p>0$, which answers a question of Toth and Zelditch \cite{TZ}.
