Abstract: For certain CM abelian varieties, there is an algebraic Hecke character such that the L-function of the abelian variety matches that of the algebraic Hecke character. This was shown by Shimura and Taniyama. 
For a Hecke character arising in this way and a positive integer n, one can ask for a Chow motive whose L-function is equal to the L-function of the n-th power of the Hecke character.  We construct such Chow motives for a particular class of algebraic Hecke characters arising from Jacobians of Weil curves. Roughly, such curves are of the form y^e = x^f + 1.
Furthermore, we discuss the field of definition of the Chow motives.  This work is joint with Laure Flapan.