After you file

The form they want you to print is called a "Voucher"

 

The form specifically identifies you as the payer...and the amount you are paying.....usually in a computer readable format that aids processing.....AND..what tax year the payment is to be applied to.   Simply sending a check doesn't do that.

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Go to your state's tax website....they usually have downloadable voucher forms there that you can fill out to include with your check.....ALSO...most state tax websites have an electronic payment link where you can arrange a direct-debit from your account instead of sending a check.

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If you are e-filing your state tax forms....DO NOT SEND a payment until your state tax file has been "Accepted".  Not just sent, not just pending, but "Accepted".  And that doesn't happen until sometime after the IRS accepts the Federal e-file.  Why?? if an e-file is rejected...then any change you make to correct the problem could also change what you owe.

____________*Answers are correct to the best of my knowledge when posted, but should not be considered to be legal or official tax advice.*