  In this article we give a sharp upper bound on the possible values of the
Euler characteristic for a minimal symplectic filling of a tight contact
structure on a lens space. This estimate is obtained by looking at the topology
of the spaces involved, extending this way what we already knew from the
universally tight case to the virtually overtwisted one. As a lower bound, we
prove that virtually overtwisted structures on lens spaces never bound Stein
rational homology balls. Then we turn our attention to covering maps: since an
overtwisted disk lifts to an overtwisted disk, all the coverings of a
universally tight structure are themselves tight. The situation is less clear
when we consider virtually overtwisted structures. By starting with such a
structure on a lens space, we know that this lifts to an overtwisted structure
on $S^3$, but what happens to all the other intermediate coverings? We give
necessary conditions for these lifts to still be tight, and deduce some
information about the fundamental groups of the possible Stein fillings of
certain virtually overtwisted structures.
