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We derive a central limit theorem for the ordered singular spectrum $\\sigma_{sing}(S_k)$ with a normal distribution as limit with explicit analytic formulas for the drift vector and the covariance matrix.\n  The main ingredient for the proof will be a oscillatory result for the spherical functions $\\phi_{i\\rho+\\lambda}$ of $(GL_n(\\b F),U_n(\\b F))$. 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