A few Galactic classical Cepheids were observed in the programmes of space missions as Coriolis, MOST and Kepler. An appealing opportunity was to detect additional nonradial modes, thus opening the possibility to perform asteroseismic studies and making the pulsational content of Galactic Cepheids more similar to that of Magellanic Clouds ones. However, only hints of cycle-to-cycle variations were found, without any strict periodicity. In this context the potential of the CoRoT exoplanetary data base was not fully exploited despite the wide area covered on the Galactic plane. Therefore, we investigated all the candidate Cepheids pointed out by the automatic classification of the CoRoT curves. At the end we could identify seven bona-fide Cepheids. The light curves were investigated to remove some instrumental effects. The frequency analysis was particularly delicate since these small effects can be enhanced by the large amplitude, resulting in the presence of significant, but spurious, peaks in the power spectrum. Indeed, the careful evaluation of a very attracting peak in the spectra of CoRoT 0102618121, allowed us to certify its spurious origin. Once that the instrumental effects were properly removed, no additional mode was detected. On the other hand, cycle-to-cycle variations of the Fourier parameters were observed, but very small and always within 3 sigma. Among the seven Cepheids, there are two Pop. I first-overtone pulsators, four Pop. I fundamental mode pulsators, and one Pop. II star. The CoRoT colours allowed us to measure that times of maximum brightness occur a little earlier (about 0.01 period) at short wavelengths than at long ones.
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