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We report a detailed CO(1-0) survey of a galaxy protocluster field at$z=2.16$, based on 475 hours of observations with the Australia TelescopeCompact Array. We constructed a large mosaic of 13 individual pointings,covering an area of 21 arcmin$^2$ and $\pm6500$ km/s range in velocity. Weobtain a robust sample of 46 CO(1-0) detections spanning $z=2.09-2.22$,constituting the largest sample of molecular gas measurements in protoclustersto date. The CO emitters show an overdensity at $z=2.12-2.21$, suggesting agalaxy super-protocluster or a protocluster connected to large-scale filamentswith ~120 cMpc size. We find that 90% CO emitters have distances $>0'.5-4'$ tothe center galaxy, indicating that small area surveys would miss the majorityof gas reservoirs in similar structures. Half of the CO emitters havevelocities larger than escape velocities, which appears gravitationally unboundto the cluster core. These unbound sources are barely found within the$R_{200}$ radius around the center, which is consistent with a picture in whichthe cluster core is collapsed while outer regions are still in formation.Compared to other protoclusters, this structure contains relatively more COemitters with relatively narrow line width and high luminosity, indicatinggalaxy mergers. We use these CO emitters to place the first constraint on theCO luminosity function and molecular gas density in an overdense environment.The amplitude of the CO luminosity function is 1.6$\pm$0.5 orders of magnitudeshigher than observed for field galaxy samples at $z\sim2$, and one order ofmagnitude higher than predictions for galaxy protoclusters from semi-analyticalSHARK models. We derive a high molecular gas density of $0.6-1.3\times10^{9}$$M_\odot$ cMpc$^{-3}$ for this structure, consistent with predictions for coldgas density of massive structures from hydro-dynamical DIANOGA simulations.

COALAS: I. ATCA CO(1-0) survey and luminosity function in the Spiderweb protocluster at z=2.16