How does this definition define a größencharakter?

Question

In Elliptic Modular Forms and Their Applications, p.89, Zagier defines a "größencharakter" $\psi_N$ on the field $K = \mathbb{Q}(i)$. This is a character on ideals. Since $\mathcal{O}_K = \mathbb{Z}[i]$ is a PID we can write every integral ideal $I\subseteq \mathcal{O}_K$ as $I = (\lambda)$ for a choice of $4$ different generators $\lambda\in\mathcal{O}_K$ (since $w_K = \lvert\mathcal{O}_K^\times\rvert=4$). Specifically, define

$$\psi_N (I) = \overline{\lambda}^N, \quad \text{ if } w_K\vert N \text{ and } I = (\lambda),$$

and set it to be $0$ for $N$ not divisible by $w_K$.

More generally, assume that $K$ is an imaginary quadratic number field with class number $h_K = 1$, and let $w_K$ denote the size of the unit group $\mathcal{O}_K^\times$. Define

$$\psi_N(I)=\overline{\lambda}^N, \quad \text{ if } w_K\vert N \text{ and } I = (\lambda).$$

Question: How can $\psi_N$ be interpreted as a größencharakter on $K$?

A größencharakter $\chi$ has 2 equivalent formulations (I believe): either as a continuous character on the idele group $\chi: \mathbb{A}_K^\times\to\mathbb{C}^\times$ with $K^\times$ in its kernel, or (modulo some ideal $\mathfrak{m}\subseteq \mathcal{O}_K$) as a character defined on the group $J^\mathfrak{m}$ of ideals coprime to $\mathfrak{m}$ such that there exist characters $$\chi_f: (\mathcal{O}_K/\mathfrak{m})^\times\to\mathbb{C}^\times, \quad \chi_\infty:\mathbf{R}^\times\to\mathbb{C}^\times$$ which determine $\chi$ on principal integral ideals: $\chi((a)) = \chi_f(a)\chi_\infty(a)$ when $a\in\mathcal{O}_K$ is coprime to $\mathfrak{m}$. Here, $\mathbf{R}$ is the $\mathbb{R}$-algebra $$\mathbf{R} = \left\{(z,\overline{z}): z\in\mathbb{C}\right\}$$ and $K\hookrightarrow\mathbf{R}$ via $a\mapsto (a, \overline{a})$. (This definition of a größencharakter can be found, for example, in Neukirch's Algebraic Number Theory, p.470.)

Ideally I would like to see how either of these could be written down explicitly for this "größencharakter" $\psi_N$, and to understand how to extend this ad-hoc definition to a größencharakter in either sense.

Answer 1

$\psi_N$ is a Größencharakter modulo $\mathfrak{m}:=\mathcal{O}_K$ in the following way. Let $\chi_f$ be the trivial character on the $1$-element group $(\mathcal{O}_K/\mathfrak{m})^\times$, and let $\chi_\infty(z,\overline{z}):=\overline{z}^N$. Then, for any nonzero $a\in\mathcal{O}_K$, we have $\psi_N((a))=\overline{a}^N=\chi_f(a)\chi_\infty(a)$, and we are done.

Note that the condition $w_K\mid N$ ensures that $\psi_N$ is well-defined: if $a,b\in\mathcal{O}_K$ are nonzero, and $(a)=(b)$, then $a=bu$ for some unit $u\in\mathcal{O}_K^\times$, and hence $a^N=(bu)^N=b^N$.