  We prove that if $\nu$ has small norm with respect to the level and the
weight, the $\nu$-th Hilbert Poincar\'e series does not vanish identically. We
also prove Selberg's identity on Kloosterman sums in the case of number fields,
which implies certain vanishing and non-vanishing relations of Hilbert
Poincar\'e series when the narrow class number is $1$. Finally, we pass to the
adelic setting and interpret the problem via Hecke operators.
