  We study fermions and hardcore bosons with long range dipolar interactions at
fractional fillings in a topological checkerboard lattice with short-range
hoppings up to next-next-nearest neighbors \cite{Neupert2011}. We consider the
case that the dipoles are aligned in the perpendicular direction by an external
field without the complication of anisotropic interaction. Using exact
diagonalization, we find clear signatures of fractional quantum Hall (FQH)
states at filling factors 1/3 and 1/5 for fermions (1/2 and 1/4 for bosons) in
the lowest Chern band with a robust spectrum gap at moderate dipolar
interaction strength. The robustness of these FQH states against long-range
interaction tail and band flatness is investigated. When the dipolar
interaction decreases, the fermionic FQH states turn into normal states, and
the bosonic 1/4-FQH state turns into a superfluid state. The bosonic 1/2-FQH
state survives even in the absence of the dipolar interaction, but vanishes
when the hard core becomes a soft core with a critical onsite repulsion. In the
thin torus limit, the static density structure factors indicates that the FQH
state turns into a commensurate charge density wave (CDW) state.
