$\mathbb{Q}$-curves and the Lebesgue-Nagell equation

In this paper, we consider the equation $$x^2 - q^{2k+1} = y^n, \qquad q \nmid x, \quad 2 \mid y,$$ for integers $x,q,k,y$ and $n$, with $k \geq 0$ and $n \geq 3$. We extend work of the first and third-named authors by finding all solutions in the cases $q= 41$ and $q = 97$. We do this by constructing a Frey-Hellegouarch $\mathbb{Q}$-curve defined over the real quadratic field $K=\mathbb{Q}(\sqrt{q})$, and using the modular method with multi-Frey techniques.