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We consider Dirichlet eigenfunctions u_\lambda, satisfying (\Delta - \lambda^2) u_\lambda = 0 S = R \cup W is the central rectangle and W as \lambda \to \infty C \lambda^{-2} mass of u_\lambda, assuming that u_\lambda-normalized; in other words, the L^2 is controlled by \lambda^2 norm in W is an o(\lambda^{-2}) quasimode we prove that the L^2 is controlled by \lambda^4 norm in W norm of u_\lambda w \vert\partial_N u\vert^2 \partial S \cap W is a smooth factor on W. These results complement recent work of Burq-Zworski which shows that the L^2 is controlled by the L^2, but adjacent to W