We present the results from a MUSE survey of twelve $zsimeq3.15$ quasars, which were selected to be much fainter (20<i<23) than in previous studies of Giant Ly$alpha$ Nebulae around the brightest quasars (16.6<i<18.7). We detect HI Ly$alpha$ nebulae around 100% of our target quasars, with emission extending to scales of at least 60 physical kpc, and up to 190 pkpc. We explore correlations between properties of the nebulae and their host quasars, with the goal of connecting variations in the properties of the illuminating QSO to the response in nebular emission. We show that the surface brightness profiles of the nebulae are similar to those of nebulae around bright quasars, but with a lower normalization. Our targeted quasars are on average 3.7 magnitudes (~30 times) fainter in UV continuum than our bright reference sample, and yet the nebulae around them are only 4.3 times fainter in mean Ly$alpha$ surface brightness, measured between 20 and 50 pkpc. We find significant correlations between the surface brightness of the nebula and the luminosity of the quasar in both UV continuum and Ly$alpha$. The latter can be interpreted as evidence for a substantial contribution from unresolved inner parts of the nebulae to the narrow components seen in the Ly$alpha$ lines of some of our faint quasars, possibly from the inner CGM or from the host galaxys ISM.
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