  In this article we construct for any prime power $q$ and odd $n \ge 5$, a new
$\mathbb{F}_{q^{2n}}$-maximal curve $\mathcal X_n$. Like the Garcia--G\"
uneri--Stichtenoth maximal curves, our curves generalize the
Giulietti--Korchm\'aros maximal curve, though in a different way. We compute
the full automorphism group of $\mathcal X_n$, yielding that it has precisely
$q(q^2-1)(q^n+1)$ automorphisms. Further, we show that unless $q=2$, the curve
$\mathcal{X}_n$ is not a Galois subcover of the Hermitian curve. Finally, we
find new values of the genus spectrum of $\mathbb{F}_{q^{2n}}$-maximal curves,
by considering some Galois subcovers of $\mathcal X_n$.
