LHS 1140 is an M dwarf known to host two known transiting planets at orbital periods of 3.77 and 24.7 days. The external planet (LHS 1140 b) is a rocky super-Earth that is located in the middle of the habitable zone of this low-mass star, placing this system at the forefront of the habitable exoplanet exploration. We further characterize this system by improving the physical and orbital properties and search for additional planetary-mass components in the system, also exploring the possibility of co-orbitals. We collected 113 new radial velocity observations with ESPRESSO over a 1.5-year time span with an average photon-noise precision of 1.07 m/s. We determine new masses with a precision of 6% for LHS 1140 b ($6.48 pm 0.46~M_{oplus}$) and 9% for LHS 1140 c ($m_c=1.78 pm 0.17~M_{oplus}$), reducing by half the previously published uncertainties. Although both planets have Earth-like bulk compositions, the internal structure analysis suggests that LHS 1140 b might be iron-enriched. In both cases, the water content is compatible to a maximum fraction of 10-12% in mass, which is equivalent to a deep ocean layer of $779 pm 650$ km for the habitable-zone planet LHS 1140 b. Our results also provide evidence for a new planet candidate in the system ($m_d= 4.8pm1.1~M_{oplus}$) on a ~78.9-day orbital period, which is detected through three independent methods. The analysis also allows us to discard other planets above 0.5 $M_{oplus}$ for periods shorter than 10 days and above 2 $M_{oplus}$ for periods up to one year. Finally, our analysis discards co-orbital planets of LHS 1140 b down to 1 $M_{oplus}$. Indications for a possible co-orbital signal in LHS 1140 c are detected in both radial velocity and photometric data, however. The new characterization of the system make it a key target for atmospheric studies of rocky worlds at different stellar irradiations
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