So long as your new computer's ethernet cable plugs into the same network as your wifi, it should work just the same. Although it sounds like it's not working for you :disappointed:.
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Here's a small checklist that might help:
- If you're on windows, make sure the network is set as a "private" network (not "public"). This means you trust connections within the network.
- Check you are connecting to the same gateway. (search that on the internet if you don't know what I mean)