Federal and state refunds come from completely separate entities, There is no rule as to which one will come first or how much time there will be between their arrivals. Some states process returns quickly and some are very slow.
STATE RETURN
Make sure your state return was accepted: https://turbotax.intuit.com/tax-tools/efile-status-lookup/
To track your state refund:
https://ttlc.intuit.com/questions/1899433-how-do-i-track-my-state-refund
Once your return is filed, TurboTax has no control over the processing of your return or the release of your refund. See the link below for tracking your state refund.
Here are a couple FAQs to track your refund:
When will I get my federal tax refund?
How do I track my state refund?
Why does Where's My Refund say "approved" but TurboTax still says "accepted"?
You would need to call the State of NY Dept of Revenue to inquire.
8:30 a.m. – 4:30 p.m.