Since 2011, the SOPHIE spectrograph has been used to search for Neptunes and super-Earths in the Northern Hemisphere. As part of this observational program, 290 radial velocity measurements of the 6.4 V magnitude star HD 158259 were obtained. Additionally, TESS photometric measurements of this target are available. We present an analysis of the SOPHIE data and compare our results with the output of the TESS pipeline. The radial velocity data, ancillary spectroscopic indices, and ground-based photometric measurements were analyzed with classical and $ell_1$ periodograms. The stellar activity was modeled as a correlated Gaussian noise and its impact on the planet detection was measured with a new technique. The SOPHIE data support the detection of five planets, each with $m sin i approx 6 M_oplus$, orbiting HD 158259 in 3.4, 5.2, 7.9, 12, and 17.4 days. Though a planetary origin is strongly favored, the 17.4 d signal is classified as a planet candidate due to a slightly lower statistical significance and to its proximity to the expected stellar rotation period. The data also present low frequency variations, most likely originating from a magnetic cycle and instrument systematics. Furthermore, the TESS pipeline reports a significant signal at 2.17 days corresponding to a planet of radius $approx 1.2 R_oplus$. A compatible signal is seen in the radial velocities, which confirms the detection of an additional planet and yields a $approx 2 M_oplus$ mass estimate. We find a system of five planets and a strong candidate near a 3:2 mean motion resonance chain orbiting HD 158259. The planets are found to be outside of the two and three body resonances.
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