  A twisted state is an important yet simple form of collective dynamics in an
oscillatory medium. Here, we describe a nontrivial type of twisted state in a
system of nonlocally coupled Stuart-Landau oscillators. The nontrivial twisted
state (NTS) is a coherent traveling wave characterized by inhomogeneous
profiles of amplitudes and phase gradients, which can be assigned a winding
number. To further investigate its properties, several methods are employed. We
perform a linear stability analysis in the continuum limit and compare the
results with Lyapunov exponents obtained in a finite-size system. The
determination of covariant Lyapunov vectors allows us to identify collective
modes. Furthermore, we show that the NTS is robust to small heterogeneities in
the natural frequencies and present a bifurcation analysis revealing that NTSs
are born/annihilated in a saddle-node bifurcation and change their stability in
Hopf bifurcations. We observe stable NTSs with winding number 1 and 2. The
latter can lose stability in a supercritical Hopf bifurcation, leading to a
modulated 2-NTS.
