Chow motives associated to certain algebraic Hecke characters

For certain CM abelian varieties $A$, there is an algebraic Hecke character
$\lambda_A$ such that $L(A, s) = L(\lambda, s)$.  If $A$ is a factor of the
Jacobian of a Weil curve (given by an equation of the form $y^e = \delta x^f + \gamma$),
we discuss a way to construct a Chow motive $M$ such that $L(M, s) = L(\lambda^n, s)$
for any positive integer $n$. Furthermore, we discuss the field of definition of $M$.