  Strong bounds are obtained for the number of automorphic forms for the group
$\Gamma_0(q) \subseteq \operatorname{Sp}(4,\mathbb{Z})$ violating the Ramanujan
conjecture at any given unramified place, which go beyond Sarnak's density
hypothesis. The proof is based on a relative trace formula of Kuznetsov type,
and best-possible bounds for certain Kloosterman sums for
$\operatorname{Sp}(4)$.
