When you go through the New York (NY) return watch for the 'credit for tax paid to another state'. Make sure the credit has been applied.
If you should happen to prepare your resident return first (NY) you should write down the income taxed by New Jersey(NJ) and the actual tax applied on the NJ return in case you need to enter the information in NY. Make sure to get the tax liability and not the refund or balance due.
If you do the NJ return first and you have entered your resident address in your tax return, then it should be automatically applied with TurboTax and you should see it as you step through the NY return.
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