I think the pot will be too big to check/shove the turn at that point. We won't be able to fold out any hands that have us beat (AQ, AJ).
$53 on flop, about $174 behind for each player
So if Villain bets 37$, and we just call, the turn potsize will be 127$, and we'll have $137 left behind. So if we check the turn to him, and he makes any sort of bet it will probably be all in and he will be pot committed even if he doesn't shove. Thus, shoving there doesn't give us any folding equity.
I think if we're going to make an aggressive play at this flop, it needs to come in one of 2 forms:
1) Check-raise all in on the flop
2) Shove all in on the turn, after we've check/called the flop.
At least at that point he will have the ability to fold some mediocre hands. And if we're going to be shoving, I prefer to do it on the flop, since we have the best odds for our draw. I think a case can be made for leading out the turn with a shove, since it looks less like a draw.
Check/calling the flop & then check/folding the turn could also be an option here, but we have a 27% shot at winning this hand against AA, and a 40% chance to win against AK & AQ. Thus, since this pot already has 50$ in it, I'm not all that comfortable throwing away the equity that we have in this pot.