Can TurboTax calculate the estimated payments for next year's state taxes?

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If you have filed your federal tax return and it has been accepted by the IRS -

When you sign onto your online account and land on the Tax Home web page, scroll down and click on Add a state. 

This will take you back to the 2021 online tax return.

To print the forms ... save the PDF of the return and choose what you want to print using the PDF reader application : 

Click on Tax Tools on the left side of the online program screen.  Then click on Print Center.  Then click on Print, save or preview this year's return.  

Choose the option Include government and TurboTax worksheets