  We prove that all mod $p^m$ singular forms of level $N$, degree $n+r$, and
$p$-rank $r$ with $n\ge r$ are congruent mod $p^m$ to linear combinations of
theta series of degree $r$ attached to quadratic forms of some level. Moreover,
we prove that, the levels of theta series are of the form
``$p\mbox{-power}\times N$''. Additionally, in some cases of mod $p$ singular
forms with smallest possible weight, we prove that the levels of theta series
should be $p$.
