period spacing variations due to a chemical gradient. This was resolved when we fit the prograde, zonal and retrograde dipole modes simultaneously, as shown in Fig.\[fig:kic9751996\_fit\]. Not only did this allow us to formally identify the $(l,m,n)$-values of the modes, it also resulted in a much higher precision for the rotation rate $f_{\rm rot} = 0.0696\pm0.0008\,d^{-1}$ and the spacing $\Delta\Pi_{l=1} =
3086\pm6\,s$ (See Fig.\[fig:kic9751996\_chi2comp\]). Interestingly, we have another indication for this derived rotation rate independently. Both in the series of the prograde modes and of the retrograde modes, we have a pulsation period which does not seem to follow the pattern, at values of 0.8 days and 0.9 days, respectively. These modes are likely trapped, which has influenced their pulsation period. When the periods of these retrograde and prograde modes are converted to their values in the corotating reference frame using the derived rotation rate, we find the pulsation periods are almost equal, which is consistent with the interpretation of trapped pulsation modes.

![\[fig:kic9751996\_chi2comp\] $\log(\chi^2_{\rm red})$ for the observed period spacing series of KIC9751996, assuming $l=1$, as a function of the asymptotic period spacing $\Delta\Pi_{l}$ and the rotation rate $f_{rot}$. The white crosses indicate the optimal found solutions. *Top:* the 