In [Elliptic Modular Forms and Their Applications](http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-74119-0_1/fulltext.pdf), 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 $K$?
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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.



How does this definition define a größencharakter?
$\psi\_N$ 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 $a^N=(bu)^N=b^N$.


