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$\begin{figure} \par\includegraphics[width=7.8cm,clip]{5015fig1.eps}\end{figure}$ Figure 1: Evolution of the frequency relative error, $E(L) = \vert\omega (L) - \omega (L^{{\rm max}})\vert / \omega (L^{{\rm max}})$ , as the spatial resolution in latitude is increased. Two modes labeled ( $\ell=1,\;n=1$ ) and ( $\ell=6,\;n=8$ ) are considered at the rotation rate $\Omega /\Omega _{\rm K} = 0.46$ with $L^{{\rm max}} = 80$ and N_r = 61.
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{5015fig2.eps}\end{figure}$ Figure 2: Evolution of the frequency relative error, $E(N_r) = \vert\omega (N_r) - \omega (N_r^{{\rm max}})\vert / \omega (N_r^{{\rm max}})$ as the resolution in the radial coordinate is increased. Two modes labeled ( $\ell=1,\;n=1$ ) and ( $\ell=6,\;n=8$ ) are considered at a rotation rate $\Omega /\Omega _{\rm K} = 0.46$ with $N_r^{{\rm max}} = 61$ and L = 62.
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In the text

$\begin{figure} \par\includegraphics[width=7.45cm,clip]{5015fig3.eps}\end{figure}$ Figure 3: Histogram of 100 frequencies obtained for 100 different values of the initial guess of the Arnoldi-Chebyshev algorithm randomly chosen in a small interval around $\omega _m = 12.0547$ . The standard deviation of this frequency distribution $\sigma = 5.6$ $\times$ 10^-6 measures the algorithm relative error on the frequency for this particular mode labeled ( $\ell=6,\;n=8$ ). The spatial resolution is N_r = 61 and L = 62 and the rotation rate is $\Omega /\Omega _{\rm K} = 0.46$ .
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In the text

$\begin{figure} \par\includegraphics[width=11cm,clip]{5015fig4.eps}\end{figure}$ Figure 4: Evolution of all the computed p-modes frequencies from $\Omega =0$ to $\Omega / \Omega _{\rm K} = 0.59$ . The frequencies have been adimensionalized by $(GM/R_{\rm p}^3)^{1/2}$ because we expect that the polar radius $R_{\rm p}$ does not change much as the rotation of the star increases. Non-rotating $\ell = (0,...,7)$ , $n= (1,...,n_{\rm max})$ p-modes have been followed by progressively increasing the rotation. This mode tracking requires special care when an avoided crossing occurs between two modes of the same equatorial parity. The figure on the left shows an overview of the frequency evolution while the two right figures display zooms to illustrate avoided crossings between the $\ell =1, \; n=6$ and $\ell =5, \; n=5$ modes and the $\ell =0, \; n=4$ and $\ell =4, \; n=3$ modes, respectively. Although the two "interacting'' modes have a mixed character near the closest frequency approach, their original properties are recovered after the crossing which enables to unambiguously follow and label the modes. This is illustrated in Fig. 5 by considering the spectra of Legendre expansion components of the $\ell =0, \; n=4$ and $\ell =4, \; n=3$ modes at the rotation rates marked by an arrow. Note that in the above figures crossings do occur between equatorially symmetric and anti-symmetric modes. In the global view, there are two examples of discontinuous frequency changes due to avoided crossing with modes which frequency is not represented on the figure. Actually, the $\ell = 8$ , n= (1,2,3) modes have been displayed in this view to avoid more discontinuous changes.
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In the text

$\begin{figure} \par\includegraphics[width=12cm,clip]{5015fig5.eps}\end{figure}$ Figure 5: Evolution of the Legendre components $C(\ell ) = {\rm max}_{n_r} \vert U(\ell ,n_r )\vert$ of the $\ell =0, \; n=4$ and $\ell =4, \; n=3$ modes during the avoided crossing shown in Fig. 4. Prior to ( $\Omega / \Omega _{\rm K} = 0.38$ ) and after ( $\Omega / \Omega _{\rm K} = 0.55$ ) the avoided crossing, the spectrum of Legendre components peaks at a given degree while, near the closest approach ( $\Omega /\Omega _{\rm K} = 0.46$ ), the double peaks of the spectra show the mixed character of the eigenmodes.
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In the text

$\begin{figure} \par\includegraphics[width=12cm,clip]{5015fig6.eps}\end{figure}$ Figure 6: Frequency spectrum of $\ell =0{-}7, \; n=1-n_{\rm max}, m=0$ modes at $\Omega =0$ ( top panel) and $\Omega / \Omega _{\rm K} = 0.59$ ( bottom panel). The degree number $\ell$ associated with the frequency is shown by the height of the vertical bar.
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In the text

$\begin{figure} \par\includegraphics[width=12cm,clip]{5015fig7.eps}\end{figure}$ Figure 7: Regularities in the frequency spacings of axisymmetric (m=0) modes. The large frequency separation between modes of consecutive order $\Delta _n = \omega _{n \ell } - \omega _{n-1 \ell }$ , the frequency separation between $\ell +2$ and $\ell$ modes, $\Delta _{2,\ell }= \omega _{n,\ell +2} - \omega _{n,\ell }$ , and the small frequency separation $\delta = \Delta _n - \Delta _{2,\ell }$ are displayed as a function of the radial order nfor four different rotation rates, a) $\Omega =0$ , b) $\Omega / \Omega _{\rm K} =0.32$ , c) $\Omega /\Omega _{\rm K} = 0.46$ and d) $\Omega / \Omega _{\rm K} = 0.59$ . We plotted the opposite of the small frequency separation $-\delta$ for clarity. Continuous lines have been drawn between frequencies of the same degree $\ell$ .
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In the text

$\begin{figure} \par\includegraphics[width=12cm,clip]{5015fig8.eps}\end{figure}$ Figure 8: Isocontours of the $\ell =4, n=4$ mode amplitude in a meridional plane as a function of the rotation a) $\Omega =0$ , b) $\Omega / \Omega _{\rm K} =0.32$ , c) $\Omega /\Omega _{\rm K} = 0.46$ and d) $\Omega / \Omega _{\rm K} = 0.59$ . The amplitude is normalized to the maximum of its absolute value. Continuous lines correspond to positive amplitudes, dashed lines to the zero amplitude and dotted lines to negative amplitudes. At zero rotation, the angular distribution is given by the $\hat{Y}^0_4 (\theta)$ Legendre polynomial while the radial distribution is characterized by the surface concentration of the amplitude and the presence of n=4 nodes in the inner part. For larger rotation rates, the largest amplitudes concentrates toward the equator. We also note that the number of radial nodes decreases along the polar radius while it increases along the equatorial radius.
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In the text

$\begin{figure} \par\includegraphics[width=12cm,clip]{5015fig9.eps}\end{figure}$ Figure 9: The mode amplitude at the surface of the polytropic model as a function of the rotation rate a) $\ell =0, n=1$ , b) $\ell =1, n=1$ , c) $\ell =6, n=8$ , d) $\ell =7, n=8$ . The amplitude is normalized to the maximum of its absolute value. While the angular distribution is given by the corresponding Legendre polynomial $\hat{Y}^0_\ell (\theta)$ in the absence of rotation, the oscillation amplitude progressively concentrates towards the equator ( $\theta = \pi /2$ ) as rotation increases.
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In the text

$\begin{figure} \par\includegraphics[width=12cm,clip]{5015fi10.eps}\end{figure}$ Figure 10: The disk-averaging factor D(i) is shown as a function of the inclination angle i for various axisymmetric modes at two different rotation rates $\Omega =0$ a) and $\Omega / \Omega _{\rm K} = 0.59$ . The degree of the modes varies from $\ell =0$ to $\ell =7$ . In the rotating case, the surface distribution also depends on the order of the mode. Two values n=1 b) and n=8 c) have been considered at $\Omega / \Omega _{\rm K} = 0.59$ .
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In the text

$\begin{figure} \par\includegraphics[width=12cm,clip]{5015fi11.eps}\end{figure}$ Figure 11: The variation of the disk-averaging factor as a function of the degree $\ell$ is shown at two fixed values of the inclination angle (i = 0 and $i= \pi /2$ ). The three curves in each figure correspond to the non-rotating case and, at $\Omega / \Omega _{\rm K} = 0.59$ , to two different orders, n=1 and n=8. The absolute value of the disk-averaging factor has been rescaled by its maximum value among the different degree considered to outline its dependency on the degree number. In sharp contrast to the non-rotating case, the disk-averaging factor at $\Omega / \Omega _{\rm K} = 0.59$ does not show a strong decrease with $\ell$ .
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{5015fi12.eps}\end{figure}$ Figure 12: The relative difference between exact frequencies and their second order perturbative approximation (second order in terms of the small parameter $\Omega / \Omega _{\rm K}$ ), namely $\delta \omega / \omega$ where $\delta \omega = \omega - \omega _{{\rm pert}}$ is displayed as a function of the rotation rate for the $\ell =0{-}2, \; n=1{-}10, \; m=0$ modes.
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{5015fig1.eps}\end{figure}$	Figure 1: Evolution of the frequency relative error, $E(L) = \vert\omega (L) - \omega (L^{{\rm max}})\vert / \omega (L^{{\rm max}})$ , as the spatial resolution in latitude is increased. Two modes labeled ( $\ell=1,\;n=1$ ) and ( $\ell=6,\;n=8$ ) are considered at the rotation rate $\Omega /\Omega _{\rm K} = 0.46$ with $L^{{\rm max}} = 80$ and N_r = 61.
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$\begin{figure} \par\includegraphics[width=7.8cm,clip]{5015fig2.eps}\end{figure}$	Figure 2: Evolution of the frequency relative error, $E(N_r) = \vert\omega (N_r) - \omega (N_r^{{\rm max}})\vert / \omega (N_r^{{\rm max}})$ as the resolution in the radial coordinate is increased. Two modes labeled ( $\ell=1,\;n=1$ ) and ( $\ell=6,\;n=8$ ) are considered at a rotation rate $\Omega /\Omega _{\rm K} = 0.46$ with $N_r^{{\rm max}} = 61$ and L = 62.
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$\begin{figure} \par\includegraphics[width=12cm,clip]{5015fig6.eps}\end{figure}$	Figure 6: Frequency spectrum of $\ell =0{-}7, \; n=1-n_{\rm max}, m=0$ modes at $\Omega =0$ ( top panel) and $\Omega / \Omega _{\rm K} = 0.59$ ( bottom panel). The degree number $\ell$ associated with the frequency is shown by the height of the vertical bar.
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$\begin{figure} \par\includegraphics[width=7.8cm,clip]{5015fi12.eps}\end{figure}$	Figure 12: The relative difference between exact frequencies and their second order perturbative approximation (second order in terms of the small parameter $\Omega / \Omega _{\rm K}$ ), namely $\delta \omega / \omega$ where $\delta \omega = \omega - \omega _{{\rm pert}}$ is displayed as a function of the rotation rate for the $\ell =0{-}2, \; n=1{-}10, \; m=0$ modes.
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