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If you expect to owe $1,000 or more in taxes for the year, beyond any withholdings, you should pay estimated quarterly taxes. The amount of estimated tax you need to pay depends on your adjusted gross income.
This link goes into more details: Quarterly Taxes
This is the IRS link that tells you how to pay the federal taxes: Estimated Taxes
How to calculate NY estimated tax payments: New York Payments
Where to pay NY payments: https://www.tax.ny.gov/pay/ind/pay-estimated-tax.htm