If you might owe a penalty under state law, you should pay early. I don't know all the state laws, I only know that for the IRS, you would not owe a penalty. If you over-pay the estimate, you will get the difference as a refund (be sure to indicate in your tax program that you made an estimated payment). As long as you don't owe a penalty, I would keep my money in my own pocket as long as possible.
Also by the way, F-1 visa holders are considered non-residents. You need to file a 1040-NR which Turbotax does not provide. Turbotax partners with Sprintax, or you can find another provider on your own.
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