  We consider the classical theta operator $\theta$ on modular forms modulo
$p^m$ and level $N$ prime to $p$ where $p$ is a prime greater than 3. Our main
result is that $\theta$ mod $p^m$ will map forms of weight $k$ to forms of
weight $k+2+2p^{m-1}(p-1)$ and that this weight is optimal in certain cases
when $m$ is at least 2. Thus, the natural expectation that $\theta$ mod $p^m$
should map to weight $k+2+p^{m-1}(p-1)$ is shown to be false.
  The primary motivation for this study is that application of the $\theta$
operator on eigenforms mod $p^m$ corresponds to twisting the attached Galois
representations with the cyclotomic character. Our construction of the
$\theta$-operator mod $p^m$ gives an explicit weight bound on the twist of a
modular mod $p^m$ Galois representation by the cyclotomic character.
