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The generic cusp is isomorphic to ${\\bf{X}}\\times ]1,+\\infty [$ with metric $ds^2=(h+dy^2)/y^{2\\delta}.$ {\\bf{X}} is a compact manifold with nonzero first Betti number equipped with the metric $h.$ For a one-form $A$ on {\\bf{M}} such that in each cusp $A$ is a non exact one-form on the boundary at infinity, we prove that the magnetic Laplacian $-\\Delta_A=(id+A)^\\star (id+A)$ satisfies the Weyl asymptotic formula with sharp remainder. 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