  Strong bounds - going beyond Sarnak's density hypothesis - are obtained for
the number of automorphic forms for the congruence subgroup Gamma_0(q) of
SL_n(Z) violating the Ramanujan conjecture at any given unramified place. The
proof is based on a relative trace formula of Kuznetsov type and best-possible
bounds for certain Kloosterman sums for GL(n). Further applications are given.
