Let $K=\mathbb{Z}(\sqrt{-5})$ be a field and $\mathcal{O}_K=\mathbb{Z}(\sqrt{-5})$ be the ring of integers. What are the ideals $\mathfrak{a} \subseteq \mathcal{O}_K=\mathbb{Z}(\sqrt{-5})$ with norm $N(\mathfrak{a})<100$? For a principal ideal $\mathfrak{a}=(a_1+a_2\sqrt{-5})$ the norm is $a_1^2+5a_2^2<100$ this region is bounded by an ellipse and so these ideals are straightforward to list. However, I know this field has class number two so this list must be incomplete.
Even a list of prime ideals might be more straightforward to write. We could list primes with $[p\mathbb{Z}(\sqrt{-5}):\mathbb{Z}(\sqrt{-5})]=p^2=100$ and some of these lattices will factor. However there are certainly other prime ideals $\mathfrak{p}\subseteq \mathbb{Z}(\sqrt{-5})$ that do not lie on the real number line $\mathbb{R}$.
On Wikipedia there's a fairly long list of Gaussian integers and their prime factorization in $\mathbb{Z}(i)$. However no such data set exists in this ring.
Best Answer
You can consider how primes split in $\mathcal{O}_K=\mathbb{Z}(\sqrt{-5})$. If we try to find the discriminant since $-5 \equiv 3\pmod 4$, $D= 4 \cdot -5=-20$, and primes not dividing $-20$ split if $\left(\frac{-20}{p}\right)=\left(\frac{-4}{p}\right)\left(\frac{p}{5}\right)=1$. From there you should be able to get a set of congruence conditions for primes that split. As far as primes that don't split, you only need consider those primes $<10$, since otherwise $N(p)=p^2>100$. Finally $2$ and $5$ both ramify so primes of $\mathcal{O}_K$ lying above them also need to be considered but should be a short, finite check from there.