  For a rational prime $p \geq 3$ we show that a $p$-ordinary modular eigenform
$f$ of weight $k\geq 2$, with $p$-adic Galois representation $\rho_f$, mod
${p^m}$ reductions $\rho_{f,m}$, and with complex multiplication (CM), is
characterized by the existence of $p$-ordinary CM companion forms $h_m$ modulo
$p^m$ for all integers $m \geq 1$ in the sense that $\rho_{f,m}\sim
\rho_{h_m,m}\otimes\chi^{k-1}$, where $\chi$ is the $p$-adic cyclotomic
character.
