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We consider a non compact, complete manifold {\bf{M}} of finite area withcuspidal ends. 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 withnonzero 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 atinfinity, we prove that the magnetic Laplacian $-\Delta_A=(id+A)^\star (id+A)$satisfies the Weyl asymptotic formula with sharp remainder. We deduce an upperbound for the counting function of the embedded eigenvalues of theLaplace-Beltrami operator $-\Delta =-\Delta_0.$

Counting function of the embedded eigenvalues for some manifold with cusps, and magnetic Laplacian