Let $\theta = (1+\sqrt{-43})/2$ and consider $\mathbb{Z}[\theta]$, a principal ideal domain, with the multiplicative map $\psi (a+b\theta)=a^2+ab+11b^2$. Show there exists an integer solution to $x^2+xy+11y^2=p$ for $p\equiv 1\pmod{43\cdot 4}$.
I've tried several approaches. One was constructing an ideal based on $\psi$ with the intention of having it be generated by some $(a,b)$ such that $\psi(a+b\theta)=p$, utilizing the principal ideal attribute of $\mathbb{Z}[\theta]$. Another was trying to more explicitly show that a solution must exist using $\psi$. However, I couldn't find a way to use the $p\equiv 1\pmod{43\cdot 4}$ property, and I didn't get far.
Pointers would be much appreciated!