A CM type for K is a choice of one out of every pair of complex conjugate embeddings of K. For each CM type on K, there is an abelian variety of that type. This is a complex abelian variety B of dimension g with an action of K (rather, its integers) such that K acts on the tangent space of B through the g chosen complex embeddings. Now you can take A to be a product of B's corresponding to whichever CM types you choose. That certainly gives you a lot of possibilities for the multiplicities.