  In this paper, we study the relation between the cocenter $\overline{\tilde
\ch_0}$ and the finite dimensional representations of an affine $0$-Hecke
algebra $\tilde \ch_0$. As a consequence, we obtain a new criterion on the
supersingular modules: a (virtual) module of $\tilde \ch_0$ is supersingular if
and only if its character vanishes on the non-supersingular part of
$\overline{\tilde \ch_0}$.
