Did you use the Desktop download program? Which version?
In the Desktop program you get one free state PROGRAM download to prepare unlimited state returns. Then each state return (including the first one) is $25 to efile. Or you can print and mail state for free.
Be aware that Deluxe desktop software comes in 2 versions--one that is a combination Federal and State version, and one that is Federal-only (no state program included). If purchased directly from TurboTax, it is the combo Federal and State version. If purchased elsewhere (Amazon, Sams, Costco, etc.) one has to be careful as to which version of Deluxe was purchased.
Which version of Deluxe do you have? If it is the combo Fed and State version, it should include a free state program download to prepare state returns for that 1 state. If it is Federal-only version of Deluxe, it will try to charge you $45 to add a state program.
In the desktop software there is a $25 fee to efile a state return. There is no state efile fee for NY state. And there is no fee to print and mail a state return.