Let U=(Uk) kЄℤ be a centered Gaussian stationary sequence satisfying some minor regularity condition. We study the asymptotic behavior of its weighted ℓ₂-norm small deviation probabilities. It is shown that (Formula Presented) using the arguments based on the spectral theory of pseudo-differential operators by M. Birman and M. Solomyak. The constant M reflects the dependence structure of U in a non-trivial way, and marks the difference with the well-studied case of the i.i.d. sequences.