It depends. If you worked in New York all year they would be taxing all of your income, first as a nonresident and then as a resident. (If you worked in NJ all year, the situation is the exact opposite). However if you moved both physically and professionally to New York, then New York does not tax all of your income. They do use all of your income, however, to determine your NY part-year resident tax.
In your situation, New York pretends that all 102000 is taxable in New York, calculates out how much taxable income you would have in New York on that income (your eligible deductions takes the amount down to $89000 of taxable income), and figures out the tax on that amount. Then the tax is prorated to the amount of income you earned in NY. (tax on $89,000, multiplied by 39,000 and divided by 102,000, which was the total gross income earned). Once prorated, the result is subtracted from the amount of NY tax that was withheld. $7 sounds like a very possible number in your case.
But NY is not taxing the income you earned in NJ while you were living there. It does use all of your income, however, to determine how much tax you must pay to NY.
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