Top-left panel: SOPHIE and HARPS-N time series superimposed with the best-fit model of the Keplerian orbit of KOI-1257.01 and the quadratic drift of the outer companion (dashed line). The best-fit residuals are shown. Bottom-left panel: phase-folded radial velocities from SOPHIE and HARPS-N after removing the quadratic drift. The best Keplerian model and its residuals are also displayed. Top-right panel: phase-folded transit light curve of KOI-1257.01 as observed during the four years of the Kepler mission. The best-fit model is superimposed to the data and the residuals from the best-fit model (in parts per thousand – ppt) are also displayed. The four different seasons of the Kepler spacecraft have different marks (see legend). Bottom-right panel: spectral energy distribution of KOI-1257 as shown in Table 1. The best stellar atmosphere model from the BT-SETTL library is also shown here together with the residuals. The best displayed models are those from Model C described in Sect. 3.6. The error bars displayed in these plots were increased quadratically by the jitter value fitted in the analyses.