No, you do not claim/get the third stimulus payment on your 2020 tax return. The third stimulus payment will be send out by the IRS separately. If you are finished with entering all you your information then I would suggest to e-file your 2020 tax return so the IRS can base the third stimulus payment on your 2020 tax return.
According to the IRS the third round of Economic Impact Payments will be based on the taxpayer's latest processed tax return from either 2020 or 2019. If the IRS has received and processed a taxpayer's 2020 return, the agency will instead make the calculation based on that return.
Please see IRS Get My Payment and How will the third stimulus impact me? for more information.
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