  We obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues
in a specific region, for $\SL_2$ over a totally real number field $F$, with
discrete subgroup of Hecke type $\Gamma_0(I)$ for a non-zero ideal $I$ in the
ring of integers of $F$. The weights are products of Fourier coefficients. This
implies in particular the existence of infinitely many cuspidal automorphic
representations with multi-eigenvalues in various regions growing to infinity.
For instance, in the quadratic case, the regions include floating boxes,
floating balls, sectors, slanted strips and products of prescribed small
intervals for all but one of the infinite places of $F$. The main tool in the
derivation is a sum formula of Kuznetsov type.
