Can be :
- Starter motor
- fault in the wire harness (short circuit)
- Starter relay
- Battery bolt not tightned
#1: happened to me (and others)
#2 : rare
#3 : I have seen a few faulty relays. So I'd check it
#4 : I guess you tightned everything when you installed the new battery, so that's not it
If it's the starter motor, take it apart and look inside. Mine was serviceable. The magnets just got loose, so I reglued all of them, cleaned everything, replaced the brushes, and reinstalled the whole thing.
If it's the relay, well ... just replace it
