  We define an analogue of the Caldero-Chapoton map (\cite{CC}) for the cluster
category of finite dimensional nilpotent representations over a cyclic quiver.
We prove that it is a cluster character (in the sense of \cite{Palu}) and
satisfies some inductive formulas for the multiplication between the
generalized cluster variables (the images of objects of the cluster category
under the map). Moreover, we construct a $\mathbb{Z}$-basis for the algebras
generated by all generalized cluster variables.
