  Let $G$ be a unimodular locally compact group. We define a property of
irreducible unitary $G$-representations $V$ which we call c-temperedness, and
which for the trivial $V$ boils down to F{\o}lner's condition (equivalent to
the trivial $V$ being tempered, i.e. to $G$ being amenable). The property of
c-temperedness is a-priori stronger than the property of temperedness.
  We conjecture that for semisimple groups over local fields temperedness
implies c-temperedness. We check the conjecture for a special class of tempered
$V$'s, as well as for all tempered $V$'s in the cases of $G := SL_2
(\mathbb{R})$ and of $G = PGL_2 (\Omega)$ for a non-Archimedean local field
$\Omega$ of characteristic $0$ and residual characteristic not $2$. We also
establish a weaker form of the conjecture, involving only $K$-finite vectors.
  In the non-Archimedean case, we give a formula expressing the character of a
tempered $V$ as an appropriately-weighted conjugation-average of a matrix
coefficient of $V$, generalising a formula of Harish-Chandra from the case when
$V$ is square-integrable.
