If it spins freely you have a a bad fan clutch.
Bmw e46 m3 fan clutch test.
Nick at pelican parts.
Open hood spin the fan by hand there should be a slight resistance and slowing it down to a dead stop.
Car must be turned off don t try to stop it while running.
Fan clutch is likely faulty.
Somewhere around the middle of temp gauge i have a 2004 bmw 325ci e46.
They can figure out what part or repair kit you need.