**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 $\mathfrak p$.

I am curious about the conductor of the above Hecke character. By Artin's reciprocity, I understand that it exists, but I am not sure how to determine it.

This MO post states that above Hecke character's conductor divides a power of $\mathfrak m$. This fact follows from class field theory, from example, from [1, VIII.5.5, page 241].

I studied the following reference that post had suggested, but I cannot figure out why the conductor divides a power of $\mathfrak{m}$. Can anyone help me with this?