Amethyst Lump is the new ore material that you can mine in Inazuma. This ore is used as the new F2P forging weapon material. You can also use Teyvat Interactive Map to see where all the Amethyst Lumps are located at.
The ore often spawns on the lower ground positions of the map. So if you see any Amethyst Lumps on the interactive map, then all you need to do is jump down and start mining it using your Claymore user, Razor or Zhongli.
With this farming route, you can get around 110 Amethyst Lumps per run. Like other mining materials, it will respawn every 72 hours (3 days) from the time of mining.
If you want the video demonstration for this farming route, you can also check the video guide at the bottom of this post.