  Relations between the symplectically harmonic cohomology and the coeffective
cohomology of a symplectic manifold are obtained. This is achieved through a
generalization of the latter, which in addition allows us to provide a
coeffective version of the filtered cohomologies introduced by C.-J. Tsai,
L.-S. Tseng and S.-T. Yau. We construct closed (simply connected) manifolds
endowed with a family of symplectic forms $\omega_t$ such that the dimensions
of these symplectic cohomology groups vary with respect to $t$. A complete
study of these cohomologies is given for 6-dimensional symplectic nilmanifolds,
and concrete examples with special cohomological properties are obtained on an
$8$-dimensional solvmanifold and on 2-step nilmanifolds in higher dimensions.
