Carlo simulation of the efficiencies in the two modes. We use the current particle data group values for the relevant $\omega$, $D^{+}$ and $D^o$ branching ratios of (88.8$\pm$0.7)% ($\omega\to \pi^+\pi^-\pi^o$), (9.0$\pm$0.6)% ($D^{+}\to K^-\pi^+\pi^+$) and (3.85$\pm$0.09)% ($D^o\to K^-\pi^+$) [@PDG]. The efficiencies listed in the table do not include these branching ratios [@spineff].

  $D^o$ Decay Mode    Fitted \# of events   Efficiency   Branching Fraction (%)
  ------------------ --------------------- ------------ ------------------------
  $K^-\pi^+$               88$\pm$14          0.064      0.41$\pm$0.07$\pm$0.04
  $K^-\pi^+\pi^+$          91$\pm$18          0.046      0.28$\pm$0.05$\pm$0.03

  : Branching Fractions for the $D\omega\pi^-$ final state[]{data-label="table:Domgpievents"}

The systematic error arises mainly from our lack of knowledge about the tracking and $\pi^o$ efficiencies. We assign errors of $\pm$2.2% on the efficiency of each charged track, and $\pm$5.4% for the $\pi^o$. The error due to the background shape is evaluated in three ways. First of all, we change the background shape by varying the fitted parameters by 1$\sigma$. This results in a change of $\pm$5.0%. Secondly, we allow the shape, $p_2$, to vary (the normalization, $p_1$, was already allowed to vary). This results in 5.5% increase in the number of events. Finally, we choose a different background function $$back'(r)=p_1 r\sqrt{1-r^2}\left(1+p_2 r + p_3 r^2 +p_4 r^3 
\right)~~~,$$ and repeat the fitting procedure. This results in a 