My 2001 ws6 had a dead battery for two weeks and I removed it from the car for about 6 days. Just put a new one on and went for a drive. Noticed it’s running a lot rougher than before and after spirited acceleration there’s a few puffs of black smoke from the exhaust and a bit of black smoke under part throttle or when I come to an idle.

I understand the pcm needs to relearn things stored in the volatile memory, so my question is are these normal things I am experiencing?