Hecke character (awaiting review)

Let $K$ be a number field with ring of integers $\Z_K$ and let $\mathfrak{m}$ be an integral ideal of $\Z_K$. Let $I(\mathfrak{m})$ be the group of fractional ideals of $\Z_K$ coprime to $\mathfrak{m}$. A Hecke character modulo $\mathfrak{m}$ is a group homomorphism $\chi:I(\mathfrak{m})\to\mathbb{C}^\times$ with the property there exists another group homomorphism $\chi_\infty:K^\times/\mathbb{Q}^\times\to\mathbb{C}^\times$ such that \[ \chi(\alpha\mathbb{Z}_K)=\chi_\infty(\alpha) \text{ for all }\alpha\in K^\times\text{ such that }\alpha\equiv1 \mod ^*\mathfrak{m}, \] where by $\alpha\equiv1\mod^*{\mathfrak{m}}$ we mean that $v_\mathfrak{p}((\alpha-1))\geq v_\mathfrak{p}(\mathfrak{m})$ for all prime ideals $\mathfrak{p}\mid\mathfrak{m}$.