New York follows the federal law in that you will get a partial exclusion on the gain of the sale of your home on both your federal and state tax returns.
TurboTax will have you enter the dates that you purchased and the dates that you sell and will give you credit for that 20K you spent on remodeling as well as any additional costs of the sale. Based on the information you have entered you'll get around a third of the exclusion or 80K or so.
That means that you will only pay tax on the amount above 385K that you receive (minus costs of the sale). Or thereabouts.
Good luck!
@naiomi82
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