  We discuss a new strategy for the computation of the Hopf-cyclic cohomology
of the Connes-Moscovici Hopf algebra $\mathcal{H}_n$. More precisely, we
introduce a multiplicative structure on the Hopf-cyclic complex of
$\mathcal{H}_n$, and we show that the van Est type characteristic homomorphism
from the Hopf-cyclic complex of $\mathcal{H}_n$ to the Gelfand-Fuks cohomology
of the Lie algebra $W_n$ of formal vector fields on $\mathbb{R}^n$ respects
this multiplicative structure. We then illustrate the machinery for $n=1$.
