Infinity type of a Hecke character

For an algebraic Hecke character, the character on archimedean ideles is a product of integral powers of embeddings and their complex conjugates. In the convention $\psi ((a))=a$ for primary $a$ over an imaginary quadratic field, the ideal character has type $(1,0)$ and the idele character has archimedean component $z^{-1}$. Distinguishing these conventions avoids inverse errors in CM Galois representations.