It depends.
If you were paid while you were a resident in New York for income that is from California sources, you could be able to claim a credit for taxes paid on any income that is taxed by both states.
However, it sounds like you only earned income in California while you lived there. When you moved to New York, you only earned income from New York sources.
If this is correct, you will only need to file two part-year resident returns. Make sure to answer the questions accordingly in the My info section of the program.
Be sure to prepare the tax return for the state you no longer live in first. Then prepare the return for the state you live in at the end of the year.
Please see this link for more direction on the steps needed within TurboTax to ensure your state returns are prepared correctly.
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