If you prepared the return on the old computer, the return was only stored on your own hard drive; you cannot access it from an online account. If you are unable to get to the file and put it on a flash drive, etc. then, sorry, but you will need to start over from scratch on that return after you have downloaded the software to the new computer.
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Are you able to get the *.tax2024 data file from the old computer? Or do you have any kind of backup on an external device or in the cloud that would have the tax file?