Negative eingenvalues of the conformal Laplacian

Let $M$ be a closed differentiable manifold of dimension at least $3$. Let ${\mathrm{\Lambda }}_0(M)$ be the minimun number of non-positive eigenvalues that the conformal Laplacian of a metric on $M$ can have. We prove that for any $k$ greater than or equal to ${\mathrm{\Lambda }}_0(M)$, there exists a Riemannian metric on $M$ such that its conformal Laplacian has exactly $k$ negative eigenvalues. Also, we discuss upper bounds for ${\mathrm{\Lambda }}_0(M)$.