Let $K=\mathbb{Q}(\sqrt{D})$ be an imaginary quadratic field of discriminant $D<0$. I want to know the image of the norm map $$ N^K_{\mathbb{Q}}:\mathcal{O}_K\to\mathbb{Z} $$ and the values of $N^K_{\mathbb{Q}}(\mathcal{O}_K)$ modulo $D$, as explicitly as possible. For example, if $D=-4$ then $N^K_{\mathbb{Q}}(\mathcal{O}_K)$ are those non-negative integers $n$ such that if $p^k||n$ and $p\equiv3(4)$, then $k$ is even, and the values of the norm modulo $4$ are $\{0,1,2\}$.
Are there similar formulations (congruence conditions, etc.) for all values of $D<0$? I'm especially interested in the values of the norm modulo $D$ as this condition occurs in something I'm working on.