No, you do not need to send any form or amend the filed tax return. The IRS will send you a tax refund for the unemployment compensation exclusion if you are eligible.
Go to this IRS website for information - https://www.irs.gov/newsroom/irs-sending-more-than-2-point-8-million-refunds-to-those-who-already-pa...
Go to this IRS website for exclusion FAQ's - https://www.irs.gov/newsroom/2020-unemployment-compensation-exclusion-faqs
