Units in residue classes

Let $K$ be a CM-field of degree $2n$. (Quadratic extension of totally real number field)

Let $\mathcal{O}_K$ be the ring of integers in $K$, and $m\geq 1$ an integer. Let $U_K$ be the group of units in $\mathcal{O}_K$.

I am interested in finding a lower bound of $T(m)$ in the following:

$$\pi_m: U_K \longrightarrow (\mathcal{O}_K/m\mathcal{O}_K)^{\times}$$

$$T(m)= |\textrm{Im}(\pi_m)|$$

I am particularly interested in lower bound of $c m^{\alpha}$ for some positive constants $c$, $\alpha$ that depend only on $K$.

Is it possible to have such bound?

Thank you.

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Sorry for not noticing that there were this: mathoverflow.net/questions/141886/… but I am still interested in finding if that bound is possible. By the way, the first comment's link contains a proof of the bound $T(m)\gg \log m$. –  i707107 Mar 7 '14 at 18:17