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Let $d \in \{-4, -8, 8\}$. We study the $8$-part of the narrow class group inthe thin families of quadratic number fields of the form$\mathbb{Q}(\sqrt{dpq})$, where $p\equiv q \equiv 1\bmod 4$ are prime numbers,and we prove new lower bounds for the proportion of narrow class groups inthese families that have an element of order $8$. In the course of our proof,we prove a general double-oscillation estimate for the quadratic residue symbolin quadratic number fields.

On the $8$-rank of narrow class groups of $\mathbb{Q}(\sqrt{-4pq})$, $\mathbb{Q}(\sqrt{-8pq})$, and $\mathbb{Q}(\sqrt{8pq})$