  The first measurements of the beam-target-helicity-asymmetries $E$ and $G$ in
the photoproduction of $\omega$-mesons off protons at the CBELSA/TAPS
experiment are reported. $E$ ($G$) was measured using circularly (linearly)
polarised photons and a longitudinally polarised target. $E$ was measured over
the photon energy range from close to threshold ($E_\gamma = 1108$~MeV) to
$E_\gamma = 2300$~MeV and $G$ at a single energy interval of $1108 < E_\gamma
<1300$~MeV. Both measurements cover the full solid angle. The observables $E$
and $G$ are highly sensitive to the contribution of baryon resonances, with $E$
acting as a helicity filter in the $s$-channel. The new results indicate
significant $s$-channel resonance contributions together with contributions
from $t$-channel exchange processes. A partial wave analysis reveals strong
contributions from the partial waves with spin-parity $J^P=3/2^+, 5/2^+$, and
$3/2^-$.
