The minimal obstruction modulus for quadratic forms

Let $Q = ax^2+bxy+cy^2$ be a primitive positive definite integral binary quadratic form with discriminant $Δ= b^2-4ac$. It is known that $Q$ admits a local obstruction; that is, there exist $k,l \in \mathbb{Z}$ such that $Q \not\equiv l \pmod k$. We study the minimal obstruction modulus $κ_Q := \min \{k \in \mathbb{Z}_{\geq 1} \mid \text{there exists } l \text{ such that } Q \not\equiv l \pmod k \}$, and we determine $κ_Q$ completely, treating the cases $Δ\equiv 0 \pmod 4$ and $Δ\equiv 1 \pmod 4$ separately. We also determine the analogous invariants for primitive ternary diagonal forms.