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 $3/2^-$.
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