  We obtain $L^p(w)$ bounds for the Carleson operator ${\mathcal C}$ in terms
of the $A_q$ constants $[w]_{A_q}$ for $1\le q\le p$. In particular, we show
that, exactly as for the Hilbert transform, $\|{\mathcal C}\|_{L^p(w)}$ is
bounded linearly by $[w]_{A_q}$ for $1\le q<p$. Our approach works in the
general context of maximally modulated Calder\'on-Zygmung operators.
