the modes with $m=0$ are indicated with dashed lines, $m=1$ with full lines and $m=2$ with the dash-dotted line.](synth_chi2_grey.pdf){width="88mm"}

![\[fig:synth\_chi2\_2d\]$\log(\chi^2_{\rm red})$ for the simulated period spacing series assuming $l=1$, as a function of the asymptotic period spacing $\Delta\Pi_{l}$ and the rotation rate $f_{rot}$. The white dot indicates the input values, while the white cross and the red boundary indicate the found solution and its 1$\sigma$-uncertainty margins respectively.](synth_chi2_2d.pdf){width="88mm"}

A slow and a fast rotator {#subsec:slowfast}
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In our sample we have one slowly rotating star, KIC9751996, for which we detected period spacing series with rotational splitting, delivering immediately the $m$-values of the modes. In order to further validate our proposed methodology, we have applied it to KIC9751996. In a first step, we only analysed the prograde period spacing pattern to test the reliability of our method. Assuming ($l$,$m$) = (1,1), this led us to find $f_{\rm rot} = 0.07\pm0.02\,d^{-1}$, which is shown in the top of Fig.\[fig:kic9751996\_chi2comp\]. However, assuming $l=1, m=0$ for the treated series, we found $f_{\rm rot} = 0.19\pm0.03\,d^{-1}$ for a similar $\chi^2$ value. The challenge in this case is that the shift and the slope in the period spacing pattern are almost negligibly small compared to the non-uniform 