  We prove Sarnak's spherical density conjecture for the principal congruence
subgroup of SL(n, Z) of arbitrary level. Applications include a complete
version of Sarnak's optimal lifting conjecture for principal congruence
subgroups of SL(n, Z), as well as a transfer of the density theorem to certain
co-compact situations. The main ingredients are new lower bounds for Whittaker
functions and strong estimates for the cardinality of ramified Kloosterman
sets.
