  Quintessential dark energy with pressure $p$ and density $\rho$ is related by
equation of state $p=\omega\rho$ with the state parameter $-1<\omega<-1/3$. The
cosmological dark energy influence on black hole spacetime are interesting and
important. In this paper, we study the Kerr-Newman-AdS solutions of the
Einstein-Maxwell equation in quintessence field around a black hole by
Newman-Janis algorithm and complex computations. From the horizon structure
equation, we obtain the expression between quintessence parameter $\alpha$ and
cosmological constant $\Lambda$ if the black hole exists two cosmological
horizon $r_{q}$ and $r_{c}$ when $\omega=-2/3$, the result is different from
rotational black hole in quintessence matter situation. Through analysis we
find that the black hole charge cannot change the value of $\alpha$. But the
black hole spin and cosmological constant are opposite. The black hole spin and
cosmological constant make the maximum value of $\alpha$ to become small. The
existence of four horizon leads seven types of extremal black holes to
constraint the parameter $\alpha$. With the state parameter $\omega$ ranging
from $-1$ to $-1/3$, the maximum value of $\alpha$ changes from $\Lambda$ to
$1$. When $\omega\rightarrow -1$, the quintessential dark energy likes
cosmological constant. The singularity of the black holes is the same with that
of Kerr black hole. We also discuss the rotation velocity of the black holes on
the equatorial plane for $\omega=-2/3,-1/2$ and $-1/3$. For small value of
$\alpha$, the rotation velocity on the equatorial plane is asymptotically flat
and it can explain the rotation curves in spiral galaxies.
