M-dwarf stars are the most common of potential exoplanet host stars in the Galaxy. It is therefore very important to understand planetary systems orbiting such stars and to determine the physical parameters of such planets with high precision. Also with the launch of the James Webb Space Telescope (JWST) the observation of atmospheric parameters of planets orbiting these stars has begun. It is therefore required to determine properties of potential targets. Two planets around the red dwarf TOI-776 were detected by TESS. The objective of our study was to use transit observations obtained by the CHEOPS space mission to improve the current precision of the planetary radii, as well as additional radial velocity (RV) data in order to improve mass estimates of the two planets. Using these quantities, we wanted to derive the bulk densities of those planets, improving the precision in earlier results, and use this information to put them in context of other exoplanetary systems involving very low mass stars. Utilizing new transit data from the CHEOPS satellite and its photometric telescope, we obtained very high precision planetary transit measurements. Interpretation of these provides updated planetary radii, along with other system parameters. A concurrent ESO large observing program using the high precision spectrograph HARPS has doubled the available radial velocity data. Calculating the power spectrum of a number of stellar activity indices we update the previously estimated stellar rotation period to a lower value. The CHEOPS data provide precise transit depths of 909 and 1177ppm translating into radii of Rb=1.798^+0.078^-0.077_R{Earth} and Rc=2.047^+0.081^-0.078_R_{Earth}, respectively. Our interpretation of the radial velocities and activity indicator time series data estimates a stellar rotation period for this early Mdwarf of ~21.1days. A further multi-dimensional Gaussian process approach confirm this new estimate. By performing a Skew-Normal (SN) fit onto the Cross Correlation Functions we extracted the RV data and the activity indicators to estimate the planetary masses, obtaining Mb=5.0^+1.6^+-1.6_M_{Earth} and Mc=6.9^+2.6^-2.5_M_{Earth}_. We improve the precision in planetary radius for TOI-776 b and c by a factor of more than two. Our data and modelling give us parameters of both bodies consistent with mini-Neptunes, albeit with a relatively high density. The stellar activity of TOI-776 is found to have increased by a factor larger than 2 since the last set of observations.