I've got axes, hatchets, machete, saws, knives, etc in my SUV at all times in NYS.
All used for trail clearing and whatnot, nothing nefarious or with any other intent. And that's kind of what the law states, intent of use.
But if it looks like a weapon you're probably going to raise some suspicion.
And I'm a long way away from NYC but I would clean my vehicle completely out before I crossed those borders.
You should be fine. In NYC I'd make sure it's in your trunk. Throw some outdoor gear in there as well. That way you can always say you are or were camping,hiking,whatever and just hadn't gotten it out of your vehicle yet. If you have it under your seat or between the seats or even with in reach its gonna throw up red flags. You may get away with it in a backpack...with hiking/camping gear in your back seat,but that's gonna depend on your record...if you have 1....and the Officer. Former NY LEO so that's the best advice I can give you.