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Down payments can be more important on new cars due to how much they depreciate when you drive off the lot (and not needing to pay for extra gap insurance if you're underwater on the loan.)
For people with borderline credit, a significant down payment can mean the difference between an approval or a denial. With your scores, I don't really think that is going to matter.
Having a lower payment (by putting more down) could be helpful if you find yourself in times of financial distress down the road (but better yet is an adequate emergency fund to cover your expenses in those times.)
If you have $15k, $20k, whatever in the bank, I might consider putting a few thousand down. If you don't, I would try to avoid it and keep your cash on hand. (Although, if I didn't I personally wouldn't be spending that much on a vehicle.)