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 attached to quadratic forms of rank r. Moreover, we prove that the levels of these theta series divide a number of the form “ \(p\text{-power }\times N\) ”. Additionally, in some cases of mod p singular forms with smallest possible weight, we prove that the levels of theta series should be p.
