Curate, connect, and discover
I wish the hoo books didn't erase the 'roman-ness' from jason completely just because he adapted to the greeks. we needed a proper, cold, calculating roman representation. hazel and frank could just as easily fit into the greeks because they were very new to camp jupiter and reyna was shipped off to the hunters.
jason was always kind of eccentric in the argo ii if you guys hadn't noticed. when jason was talking in praise about diocletian(?) hazel looked at him in horror because he wasn't supposed to be saying good things about him. but the romans were supposed to be like that. they were supposed to take in the beneficial aspects of even bad people, they didn't think about feelings. they were the "we see good in everyone but only because we need them for our benefit".
it would take a million years for jason to trust someone completely, hot take but he would take longer than nico to warm up to someone. nico is just more open about not trusting someone, but jason would hide it better. annabeth said jason had this unnerving vibe about him and neither she or hazel could get a read on him. he would be calculating your every move and would read so much into what you say, because he's a roman by blood, by virtue.
and jason would take betrayals HORRIBLY, arguably he'd take it the hardest, it isn't easy to anger him but a broken promise, a broken trust, drives him absolutely insane.