3 Data analysis

3.1 Preliminaries

The data analysis involves the following preliminary steps:

1.: The usual correction of the frames for instrumental effects were made by the observers, following the routines in use at each observatory.
2.: The preparation of each frame for measurement was made in the ESO-MIDAS environment, using a procedure summarized by Giudicelli & Michard (1993). It involves the elimination of significant parasitic objects, the extraction of a suitable stellar image (if any) for the derivation of the PSF, the evaluation of the sky background, the calibration by comparison with the results of aperture photometry (see Sect. 3.3 below); a cosmetic treatment against cosmic ray peaks and anomalous negative pixels, and reduction of the frame to the field deemed necessary. If the S/N ratio is adequate for the derivation of 2D colour maps, the above treatment may be preceded or completed by the mutual alignement of the frames against the one taken as geometrical reference, possibly using the images of stars to improve the alignment. The output of these preparations are "clean'' frames for the galaxy and the PSF, with a number of useful parameters available in the frames "descriptors''.
3.: Since the colours will be measured along arcs of isophotes, it is a necessary step to obtain a set of isophotes in tabular form. The contours are described by the well known representation first proposed by Carter (1978). For details about our implementation of Carter's representation, see Michard & Marchal (1994). Only one set of isophotal contours is used to compare two frames and get local colours, but the comparison of Carter's parameters for the two frames may be revealing, as noted for instance by Goodfroij et al. (1994a).

In previous survey of elliptical galaxies, one was often satisfied with measuring a local reference colour, plus a colour gradient, such as C₁-C₂ and d(C₁-C₂)/d $\log r$ , where r is the equivalent isophotal radius defined above. In general, however, two parameters are far from sufficient to describe the colour distribution in an early-type galaxy. There is no a priori physical reason for the logarithmic colour gradient to remain constant throughout the measured range of radii: it does not, even for a "pure spheroid'', if one is able to extend the measurements, either near the centre or outside the central body of the galaxy. There are also significant differences between the gradients in the disk and spheroid of S0's (and diE?), which translate into differences between the major and minor axis gradients for inclined objects. The presence of dust leads to various appearances: local features can sometimes be avoided in order to get more significant colour gradients. Dust layers in disks lead to minor axis asymmetries in light and colours (MS93), while diffuse dust in spheroids will modify apparent light and colour gradients (see the calculations by Witt et al. 1992). It is therefore useful to provide colour data at the relevant level of details, the ultimate being quantitative 2D colour maps.

3.2.1 1D colour measurements

To provide a good insight into the colour properties of a given object without necessarily ressorting to the 2D colour maps, we have chosen to measure both the radial and azimuthal colour distributions. We use ad hoc computer programmes, involving as input the two frames to be compared and the table of isophotal contours for one of these: the set of contours is used to locate corresponding points in the two frames. Note that for this particular purpose "symmetrized'' contours are used, retaining only the even cosine harmonics of their representation.

To get the radial distributions, local colours are averaged along arcs of the tabulated isophotes, of length 45 degrees in the eccentric anomaly $\omega$ of Carter's Reference Ellipse. Both major and minor axes, and both halves of the two axes, are measured separately. Note that, since the isophotes and isochromes are very much alike in E-S0 galaxies, averaging the colours along moderate intervals in $\omega$ of the isophotal contours, is a technically justified way to improve the S/N ratio.

To get the azimuthal distribution, local colours are averaged inside two concentric isophotes sufficiently apart to improve the S/N ratio, while the azimuthal resolution is kept to 120 points, or 3 degrees in $\omega$ . Note that evenly distant points in $\omega$ are not so in position angle from the galaxian centre: they are indeed more closely packed near the tips of the major axis. Suitable software is also available to get the azimuthal colour distribution as a function of the PA itself.

Sample outputs of the above measurements are presented in Figs. 1 and 2 for NGC 4125, a galaxy with a strong dust pattern and an exceptionally large colour gradient. Then the radial B-R profiles are quite different for the 4 mesured arcs, although averaging in rather large azimuthal domains reduces such differences. For the azimuthal profiles the averaging is performed in radial domains as noted in the figure labels, and more detail of the true B-R distributions are kept. A 2D map of the B-R distribution for the sample galaxy NGC 4125 is shown in Fig. A14 and should be compared with the 1D graphs: it preserves of course more details, but the 1D graphs are certainly useful to appreciate the significance against noise of some features of the distributions.

$\begin{figure} \resizebox {\hsize}{!}{\includegraphics{ds1642f01.eps}}\end{figure}$

Figure 1: Sample of a measured radial B-R distribution, i.e. for NGC 4125. Open circles: eastwards majA. Filled circles: westwards majA. Open squares: southwards minA. Filled squares: northwards minA. The differences are due to an important dust pattern, the data being smoothed by averaging along arcs of isophotes as explained in the text. The upper label gives the code of the B-frame and the corresponding seeing FWHM

$\begin{figure} \resizebox {\hsize}{!}{\includegraphics{ds1642f02.eps}}\end{figure}$

Figure 2: Sample of a measured azimuthal B-R distribution, i.e. for NGC 4125. Four successive "rings'' (as defined in 3.2.1), limited with the isophotes of radii 2.0, 3.2, 5.0, 7.9 and 12.6 arcsec are measured. Abscissae: PA along the isophotes counted counterclokwise from one of the majA, here the one pointing westwards. Ordinate: B-R, the scale for each ring being adjusted as needed. The large variations are due to the dust pattern. The upper label gives the code of the B frame and the corresponding seeing FWHM

3.2.2 2D colour maps

The above 1D graphs will indicate if the colour distribution follows the SuBr distribution or presents significant deviations, due to dust or population variations. In this case it is interesting to consider 2D colour maps. We have chosen to produce colour maps strictly consistent with the classic astrophysical definition of the colour as a difference of magnitudes. This requires a precise alignment of the two frames, which is not always easily achieved when there are no suitable sets of stars. Very often, one can only put in coincidence the centres of the studied galaxy in the two images... but this centre may be influenced by colour features (and of course by noise).

A number of techniques have been experimented to reduce the noise in colour maps, or more exactly to increase the range of SuBr where this noise remains acceptable, including the one introduced long ago by Sparks et al. (1985): they replace the redder of the pair of frames by a synthetic image built from a set of isophotal contours. It was concluded however that the improvement was not worth the extra work, since the frames studied here are useful only to measure the innermost range of galaxian colours.

3.2.3 1D and 2D asymmetries

Besides 2D colour maps, we have also considered in some cases, 1D graphs of SuBr asymmetries as in MS93, and also 2D maps of asymmetries, obtained by comparing a given image with a model of the same, calculated from the corresponding file of isophotal contours: one may use a model with elliptical isophotes, as in van Dokkum & Franx (1995), or keep the even cosine Carter's coefficients e₄, e₆, ... The model then preserves the diskyness, or boxyness, of the true contours. To get asymmetry maps, we also impose a unique centre and constant orientation to the model isophotes. Such maps have been produced in such cases where the asymmetries were thought to bring useful hints about the dust distribution, in complement to colour maps. By analogy with the findings of MS93, for S0's, it was supposed that dust concentrated in the disk of diE's, might give asymmetries in their inner bulge light: this is indeed the case for a few diE's.

Remark: It should be realized that fitting ellipses produces isophotes that are "interlaced'' with the real ones, so that differences between the true image and the model necessarily compensate "dark residuals'' (due to extinction?), by nearby "bright residuals''. Conversely, the bright residuals due to an embedded disk are compensated by dark residuals that should not be mistaken for absorption markings. In order to get reliable "extinction maps'', rather elaborate procedures are necessary, as described by Goudfrooij et al. (1994c).

3.3 Calibrations and colour corrections

The frames have been calibrated mostly from the UBVRI aperture photometry by Poulain (1988), and Poulain & Nieto (1994) where the R band is in Cousins's system. These data are available for 30 objects in the sample. For the others, the catalogues of UBV aperture photometry by Longo & De Vaucouleurs (1983, 1985) were used. The R photometry was obtained from a tight correlation between the observed B-V and V-R from Poulain (1988). For two galaxies, i.e. NGC 3613 and 4649, the available field was too small for calibration with existing aperture photometry: the first was calibrated by fitting to calibrated wide-field frames from the Observatoire de Haute-Provence; the second by fitting to the B and R data from Peletier et al. (1990) or PDIDC. As regards the Pic du Midi subset of 6 objects, 3 were calibrated from Poulain's photometry. No significant difference with the CFHT bulk of data is expected.

Poulain's aperture photometry is accurate enough to provide good tests of the sky background. If the sky can be measured with sufficient accuracy from a corner of the frame, the derived photometric zeropoint does not show systematic changes in the various apertures. If not, as it was of course the case for large galaxies, the assumed background value was varied until the test yielded satisfactory results.

The B-R colours collected in Table 6 have been corrected for Galactic reddening and K-effect. It is to be noted that graphical data are not corrected. The B-V colour excesses were derived from the B extinction values given in the Third Reference Catalog of Bright Galaxies, or RC3, by de Vaucouleurs et al. (1991). From Rieke and Lebofsky (1985), we found the galactic colour excess in B-R to be 1.75 larger than E(B-V). For the K correction the RC3 precepts were followed. and the radial velocities taken from this same source. The proper coefficient was taken from Frei & Gunn (1994). The work of Fukugita et al. (1995) was also considered.

The resulting CFHT colour system has been compared with the one of PDIDC, using the observed colours at the r=10 arcsec contours for 17 objects in common. Note that PDIDC calibrated 30 of their 39 galaxies from Burstein et al. (1987) photometry. A zero point difference for PDIDC-Us of 0.08 is found, with $\sigma=0.04$ .

Remark: In the comparison with PDIDC, the galaxy NGC 2768 was neglected: it is found much bluer by PDIDC than here, at $4\sigma$ of the above mean difference.

3.4 Errors due to"differential seeing'' and their correction

By differential seeing, we mean the fact that colour measurements involve two frames obtained with different seeing. As the usually encountered PSF's have terribly large effects upon central SuBr distributions, the difference of the two PSF's will lead to large errors in colours. These errors have been discussed by Vigroux et al. (1988), Franx et al. (1989), Peletier et al. (1990). The later authors derived a cutoff radius for each observation and discarded colours measured inside this limit: in nearly all cases this cutoff is larger than 3 arcsec, although the conditions adopted in its definition cannot be considered as very stringent.

Since we are interested in inner colour distributions, i.e. well inside the cutoff radii of PDIDC, we tried to get approximate corrections for the effects of differential seeing. The essential step of the corrections is to find a function FC, such as the convolution of the sharper of the two PSF's with FC will reproduce the other one. After convolution of the sharper frame with FC the colour distribution will be obtained with the resolution allowed by the worse of the two frames! Another possibility would be to deconvolve the more blurred of the frames by FC, in order to match it with the sharper one. Deconvolution artefacts of the kind described by Michard (1996), might be small enough in this case, because FC is expected to be much narrower than the actual PSF. Of these two possibilities the first one has been used in practice, because it is time saving.

The derivation of FC is of course dependent upon the availability of a "good'' star in the two frames. One can consider several cases, depending upon the S/N of the stellar images and their actual geometry.

1.: If the two PSF's can be approximated by Gaussians with circular symmetry, FC is simply another such Gaussian (case C1GR).
2.: If the PSF's have important wings, still with circular symmetry, FC can be better approximated by the sum of two Gaussians, intended respectively to match the core and wings of the PSF's (case C2GR).
3.: If the PSF's are elongated, often due to unequal guiding errors in $\alpha$ and $\delta$ , FC can be better approximated by a Gaussian with different $\sigma$ values in x and y (case C1GXY). Eventually the sum of two Gaussians might be considered (case C2GXY).

These various cases can be implemented by ad hoc MIDAS procedures. Very often however the S/N of available stellar images in our small field frames does not allow much refinements in the derivation of FC, and one has to be satisfied to use the C1GR approximation.

Several experiments have been made to ascertain the effects of differential seeing and the success of the above correction techniques. Part of such experiments were made with model galaxies and model PSF's. The model object, assumed colourless, was convolved with two different PSF's, a sharper one P1 and a poorer P2. The corresponding colour profiles C₁-C₂ were evaluated, and then the above corrections techniques applied. Of course, P1 and P2 are not simple Gaussians, since in this case an exact correction is readily obtained. Figures 3 to 5 present the results of three such experiments.

$\begin{figure} \resizebox {\hsize}{!}{\includegraphics{ds1642f03.eps}}\end{figure}$

Figure 3: Experiment showing a pseudo-colour profile induced by differential seeing, and its approximate correction. Abscissae: log of isophotal mean radius r in arcsec. Ordinates: Colour in magnitude, with circles for the majA and crosses for the minA. The model galaxy is a circular r^1/4 bulge (slightly modified), of FWHM = 0.60 arcsec. The PSF's are sum of Gaussians. Sharper one DW31: FWHM = 0.51 arcsec with faint wings. Broader one DW43: FWHM = 0.79 arcsec with strong wings. Upper curve: Colour profile for convolved frames of the model. Lower curve: Colour profile after PSF matching with a single circular gausian (case CG1R). The correction is quite successful

$\begin{figure} \resizebox {\hsize}{!}{\includegraphics{ds1642f04.eps}}\end{figure}$

Figure 4: Experiment showing pseudo-colour profiles induced by differential seeing, and their approximate correction. Abscissae: log of isophotal mean radius r in arcsec. Ordinates: Colour in magnitude, with circles for the majA and crosses for the minA. The model galaxy is akin to a flat disky E or S0, with a slightly modified r^1/4 bulge of ellipticity $\epsilon=0.5$ , plus a disk with $\epsilon=0.74$ .The PSF's are sum of Gaussians, i.e. the same as in Fig. 3. Upper curve: Colour profiles for convolved frames of the model. Lower curve: Colour profiles after PSF matching with a single circular gausian (case CG1R). The differential seeing results in a blue light excess on the minA, extending up to 3-4 times the broad PSF FWHM. Again the correction is successful

$\begin{figure} \resizebox {\hsize}{!}{\includegraphics{ds1642f05.eps}}\end{figure}$

Figure 5: Experiment showing pseudo-colour profiles induced by differential seeing, and their approximate correction. Abscissae: log of isophotal mean radius r in arcsec. Ordinates: Colour in magnitude, with circles for the majA and crosses for the minA. The model galaxy is the flat object considered in Fig. 4. The sharper PSF is the same as in the previous figures, but now the broader PSF DWY2 has extended wings, elongated in the direction of the minA of the galaxy. FWHM's: 0.67 arcsec along PSF minA; 0.82 arcsec along PSF majA. Upper curve: Colour profiles for convolved frames of the model. Intermediate curve: Colour profiles after PSF matching with the best circular Gaussian. Lower curve: Colour profiles after PSF matching with the best elongated Gaussian. The correction whith a circular Gaussian is now unsufficient, but it is successful with a properly elongated Gaussian

Other tests were made on real galaxies which have been observed twice in different nights and seeing conditions. Then it is possible to measure the pseudo-colours B-B and R-R resulting from the corresponding frames: this illustrates the effects of differential seeing, and the success of its correction in actual observations, that is in the presence of noise. The Figs. 6 and 7 give examples of these tests, which were applied to 12 pairs of frames. Finally, Fig. 8 shows how the central colour profiles will vary with slight changes in the adopted PSF matching function FC (case C1GR). The central red peak in the "observed'' colour of the test object NGC 4473 is lessened by the adopted correction, i.e. with a correcting Gaussian of $\sigma=0.2$ arcsec. It becomes a blue feature with $\sigma=0.4$ .

$\begin{figure} \resizebox {\hsize}{!}{\includegraphics{ds1642f06.eps}}\end{figure}$

Figure 6: Experimental correction of the colour profile of a real galaxy, i.e. NGC 3377 for the artefacts of differential seeing. Here are considered two frames taken in the same R passband with measured PSF's of 0.59 and 0.66 arcsec FWHM. Abscissae: log of isophotal mean radius r in arcsec. Ordinates: Colour in magnitude, with circles for the majA and crosses for the minA. The uncorrected R-R colour (upper graph) shows a central red peak and minA blueing as for the model of Fig. 4. After correction (lower graph) the R-R colour becomes flat except for noise fluctuations

$\begin{figure} \resizebox {\hsize}{!}{\includegraphics{ds1642f07.eps}}\end{figure}$

Figure 7: Experimental correction of the colour profile of a real galaxy, i.e. NGC 3115 for the artefacts of differential seeing. Here are considered two frames taken in the same R passband with measured PSF's of 0.55 and 0.91 arcsec FWHM. Abscissae: log of isophotal mean radius r in arcsec. Ordinates: Colour in magnitude, with circles for the majA and crosses for the minA. Due to widely different PSF's, the uncorrected R-R colour (upper graph) shows a strong central red peak and minA blueing as for the model of Fig. 4. After correction (lower graph) the R-R colour becomes much flatter, except for a local slight bump on the minA

$\begin{figure} \resizebox {\hsize}{!}{\includegraphics{ds1642f08.eps}}\end{figure}$

Figure 8: Corrections to the radial colour distributions for NGC 4473. The frames had seeing FWHM of 1.19 arcsec in B and 1.05 in R, poorly measured on a faint star. Abscissae: log of isophotal mean radius r in arcsec. Ordinates: Colour in magnitude, with circles for the majA and crosses for the minA. Upper graph: uncorrected results. Intermediate: adopted correction with a Gaussian of $\sigma$ = 0.2 arcsec. Lower graph: "overcorrection'' with $\sigma$ = 0.37. The central red peak progressively turns out into a blue feature

From the tests here described we draw the following conclusions:

1.: the errors in peak core colours due to differential seeing may reach several tenths of magnitude, also at the relatively good seeing conditions of the CFHT.
2.: such errors, at the level of 0.02 mag, extend only up to a radius of twice the FWHM of the PSF (the worse one) for a roundish object and circular PSF's. But the effects are much worse for an elongated PSF "crossing" the minA of a flattened galaxy (see Fig. 5). In this case the geometry of the inner isochromes may be seriously modified.
3.: the errors here discussed may be much reduced by matching the PSF's, as described above. The improvement is limited by difficulties in getting well defined PSF's from the noisy images of faint stars.

3.5 Comparison of results for multiply observed objects

For several of the sample galaxies, more than one frame of suitable S/N ratio are available in one or both colours (neglecting very short core exposures!). Such multiple observations may be used to evaluate part of the errors involved in the present work. Two different approaches were found useful.

3.5.1 Pseudo-colours from pairs of frames with the same filter

The derivation of pseudo-colours B-B or R-R gives useful information about errors of various origins. The following cases should be distinguished in these experiments:

1.: For frames where the galaxy is located at widely different positions within the instrumental field, or taken with different instrumentation, the errors in background level or flat-field trends will be uncorrelated. The calibrations may also differ, if the number of aperture photometry results is not the same for the two frames. It was found that large residuals may occur under these circumstances. It is therefore advisable to derive colours from pairs of frames taken in succession during the same night and without large offsets of the object within the field. This was the usual practice for the observers who collected the presently used material.
2.: For frames taken with the same instrument and with the galaxy at nearly the same location on the CCD target, the residuals in pseudo-colours are due essentially to inaccuracies in the PSF matching. Other errors, such as resulting from the background estimate or residual trends in flat-fielding, will be correlated in the treatment of such parent frames. Results for these cases have been considered above (see Figs. 6 and 7).

3.5.2 Comparisons of colour distributions from different pairs of frames

Such comparisons could be achieved for 8 galaxies: two were observed on two consecutive nights, the second with better seeing; four were re-observed with the HRCam in the hope to get better resolution. For the other there was some duplication of the data in a single night, a case of limited interest because the differences are due mainly to errors in PSF matching, already discussed above. Here are discussed only the cases where the errors in the B-R distributions are largely independent, except for the errors of calibration. Table 5 presents the results of these comparisons, using ad hoc parameters.

**Table 5:** Comparison of B-R data from multiple observations. Fr: code for the pair of frames, for reference to Table 1. W_B: *FWHM* in B. W_R: *FWHM* in R. C₀: B-R at core centre. C₁: B-R at radius r=1 arcsec. C₃: B-R at radius r=3 arcsec. Gr: logarithmic B-R gradient for r> 3 arcsec. The colours are here uncorrected. Nt: Notes to Table 3 (a) No star; mean *FWHM*'s for the night. (b) Galaxy near the edge of the frame to get a star in. (c) HRCam observations
$\begin{tabular} {lllllllll} \hline NGC & Fr & $W_B$\space & $W_R$\space & $C_0$\... ... 0.89 & 0.79 & 1.66 & 1.57 & 1.48 & $-0.07$\space & (c) \\ \hline\end{tabular}$

The results of the present experiments are summarized below, both from pseudo-colours and from multiple colour observations:

1.: Errors due to imperfect PSF matching are restricted to a radius roughly equal to the worse PSF FWHM. An estimate of random errors upon the central core colour is 0.03. For strongly flattened objects errors upon the minA colour profile may occur at the same amplitude (see the case of NGC 3115, in Fig. 4).
2.: Spurious colour patches at an amplitude of 0.03 may occur due to unsufficient S/N.
3.: Rather large errors may develop near the limits of the available field. This is due to poor background estimates: these may be worse than in classical observations, where the sky light is effectively registered on the frame due to adequate field of view and generous exposure. As a result the logarithmic gradients of B-R are quite uncertain. The experiment summarized in Table 5 point to a mean difference of 0.05 between two measures of the gradient for the same object! Similarly the mean difference between two measures of the colour at $r_{\rm e}/2.5$ is 0.04. The mean errors upon a single measurement will be slightly smaller.

3.6 Classification of central SuBr profiles

It has been shown by Nieto et al. (1991a), and more recently by Jaffe et al. (1994) and Lauer et al. (1995), that the "cores" of E-galaxies can be sorted out in two types, here termed flat topped core and sharp peak, or respectively ftc and shp. This corresponds to Type I and Type II in the notation of Jaffe et al., or "core-like profile" against "power-law profile" in Lauer et al. wording.

In view of a comparison of central colour profiles with the types of central SuBr profiles, it was necessary to supplement the lists of "core" types available from the quoted papers. For this purpose, the R frames were deconvolved by Lucy's technique, as implemented in the MIDAS software, using 27 iterations. Then three parameters were examined: change of peak SuBr between the original and deconvolved frames, or equivalently the ratio of the FWHM's in the original and deconvolved frames, and finally the FWHM of the deconvolved core. These three parameters indeed show a bimodal distribution. For the first two, this corresponds to the fact that ftc profiles are resolved, or nearly so, at the CFHT resolution, while the shp profiles remain quite unresolved. The last one is less dependent upon the actual PSF: it would perhaps converge towards an exact galaxian property if the number of iterations was varied in relation with the frame resolution... and if the PSF were perfectly accurate.

**Table 6:** Isophotal colours at selected radii and gradients. $M_{\rm T}$ : Absolute magnitude in B from Michard & Marchal (1994), or derived accordingly. log $r_{\rm e}$ : Logarithm of the effective isophotal radius in arcsec from same source. $\Delta(B-R)$ ; Corrections to the observed B-R for galactic extinction and K-effect. CP: ftc for a flat topped core, shp for a sharp peak. C₀: Peak central corrected B-R colour. $\Delta C_{0,3}$ colour difference between centre and radius r=3 arcsec. C₁: Corrected B-R at $r_{\rm e}/10$ . It is uncertain, or not measured, if $r_{\rm e}<10$ .C₂: Corrected B-R at $r_{\rm e}/2.5$ . It involves an extrapolation for large objects. *G₁₂*: Outer logarithmic gradient for r>3 arcsec. Code: Dust pattern importance index *DPII* and code dd for "dust in disk". A colon: refers to an uncertain result. Notes to Table 6: (a) NGC 0636: small blue dot in core: representative core colour interpolated. (b) NGC 2974: peak B-R in patch 1.7 arcsec NW of centre. (c) NGC 3156: Very anomalous colour distribution (see Fig. 11) (d) NGC 4125: Extraordinary gradient (see also Goudfrooij et al. 1994a) (e) NGC 4742: Very anomalous colour distribution (see Fig. 12) (f) NGC 5322: Inner small disk, with some reddening, isolated from main boxy body. (g) NGC 5845: Inner small disk, with some reddening, isolated from main disky body
$\begin{tabular} {lllllllllllll} \hline NGC~& Type & $-M_{\rm T}$\space & log$r_{... ... & 0.08 & shp & 1.65 & 0.05 & - & 1.59: & $-$.04: & 0 & - \\ \hline\end{tabular}$

The present classification of central profiles, given in Table 6, shows perfect agreement with the one of Faber et al. (1997) for 15 galaxies in common. From the graphs of Byun et al. (1996), there are a few profiles intermediate between the typical "core-like'' and "power-law'' cases. Similarly our classification gives uncertain results for NGC 4125, 5322, 5576, 5638. Not that the few objects observed at the TBL could not be classified.

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