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tokens: of photometric long-term cycles in M dwarfs). Fig. 10 GLS periodograms of CARMENES activity indicators of G 264–012 in the VIS ( top panels ) and NIR ( bottom panels ) channels. Horizontal FAP lines and vertical signal indications are the same as in Fig. 3 . Since the CARMENES spectral range does not include the Ca II HK lines, we carried out an analysis of all available archival data covering this spectral region, namely the HARPS and HIRES spectra used in the RV analysis of this work (Sect. 4.1.2 ), 1 spectrum acquired by FEROS ( Kaufer et al. 1999 ), 3 spectra from UVES ( Dekker et al. 2000 ), and 10 from XSHOOTER ( Vernet et al. 2011 ). In order to extract from spectra acquired by all five instruments in a uniform manner, we rectified the spectra with a grid of PHOENIX spectra ( Husser et al. 2013 ) by extracting a set of narrow band passes around both the H and K lines. The approach will be described in detail in an upcoming publication (Perdelwitz et al., in prep.). The top panel of Fig. 12 shows the resulting time series. The smaller periodicity in the bottom panel of Fig. 12 , with a period of P = 36.6 ± 1.9 d, where the error was determined via a Monte Carlo approach, is in agreement with the stellar rotation period derived from Kepler K2 photometry (Sect. 3.1 ). Fig. 11 GLS periodograms of CARMENES VIS, HARPS, and HIRES activity indicators for Gl 393. Horizontal FAP lines and vertical signal indications are the same as in Fig. 4 . Fig. 12 Top panel : time series of Gl 393. The source of the archival data is denoted by the colour coding. Bottom panel : generalised Lomb-Scargle periodogram of the time series of Gl 393. The dashed vertical lines denote the periods of the activity cycle ( P ≈ 1226 d) and stellar rotation ( P ≈ 36.6 d) determined from the time series. FAP levels of 10 −3 and 10 −9 are displayed as horizontal lines. 4.5 Gaussian process analysis of RV data 4.5.1 G 264–012 All indicators gathered (Sect. 4.2 – 4.4 ) point towards activity as origin for the RV signal found at a period of P d = 92.70 d in G 264–012. There is, however, no indicator contradicting the planetary nature of the other two signals, P b = 2.30 d and P c = 8.05 d. In order to accurately determine their orbital parameters and masses, we account for stellar activity in our models. We choose a model with two circular Keplerian orbits (no evidence for eccentricity was found in these two very-short period planets, whose orbits should be almost certainly already circularised, see Sect. 4.1 ) for the planets combined with a Gaussian process (quasi-periodic kernel; see Eq. ( 1 )) for the activity signal. We present the fits in Fig. 13 , a summary of all parameters used in the model in Table 5 and the MCMC posterior distribution plot in Fig. A.1 . When subtracting the model from the VIS RV data we derived an rms of 2.22 m s −1 , which is consistent with the errorbars. Fig. 13 G 264–012. Panel a : GP fit (black solid line) and 1-sigma uncertainties (grey