Notation. Let $n$ be a positive integer, let $\mu_n\subseteq \mathbb C$ be the set of $N$-th roots of unity, let $K$ be a number field containing $\mu_n$, let $R$ be the ring of integers of $K$, let $a\in R\setminus\{0\}$, let $\mathfrak m\subset R$ be the ideal generated by $a$ and $n$, let $\mathcal I_\mathfrak m$ be the set of fractional ideal of $K$ coprime with $\mathfrak m$.
The Power Residue Symbol is the multiplicative character $\bigl(\frac a \cdot\bigr): \mathcal I_\mathfrak m\to \mu_n$ that satistfies $\bigl(\frac a {\mathfrak p}\bigr)\equiv a^{\frac{N\mathfrak p-1}{n}} \mod {\mathfrak p}$ for every prime ideal $\mathfrak p\in\mathcal I_\mathcal m$, where $N\mathfrak p\in\mathbb N$ is the norm of $\mathfrak p$.
Fact. If I am not mistaken, the power residue symbol is a unitary Hecke character (aka Grossencharakter) of $K$, with trivial infinity type, with finite order, and with conductor dividing a power of $\mathfrak m$. This fact follows from class field theory, from example, from [1, VIII.5.5, page 241].
Question. Is there a citable reference in which this fact is stated more explicitly? I looked also in [2] and in a plethora of other places, but I was not successful.
[1]: J. Milne. Class field theory (v4.02), 2013. Available at www.jmilne.org/math/
[2]: Neukirch, Jürgen. Class field theory. Vol. 280. Berlin: Springer, 1986.
