  In this paper, we introduce the Newton decomposition on a connected reductive
$p$-adic group $G$. Based on it we give a nice decomposition of the cocenter of
its Hecke algebra. Here we consider both the ordinary cocenter associated to
the usual conjugation action on $G$ and the twisted cocenter arising from the
theory of twisted endoscopy.
  We give Iwahori-Matsumoto type generators on the Newton components of the
cocenter. Based on it, we prove a generalization of Howe's conjecture on the
restriction of (both ordinary and twisted) invariant distributions. Finally we
give an explicit description of the structure of the rigid cocenter.
