Let F be a totally real number field of dimension d, and let OF be its ring of integers. If q is an
ideal in OF let Γ= Γ0 (q) denote the congruence subgroup of Hecke type of the Hilbert
modular group. In the present paper, we derive density results for cuspidal automorphic
representations of G= SL2 (R) d in L2 (Γ (q)\G). The main result, Theorem 3.3, implies that
there are infinitely many cuspidal automorphic representations ϖ=⊗ dj= 1ϖj, even if we
restrict some components ϖj. In particular, let Λ be the set of eigenvalue vectors λϖ=(λϖ1 …

We obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific
region, for $\mathrm {SL} _2 $ over a totally real number field $ F $, with a 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 …