Cessna Pilot’s Assn has good (and correct) info on nosewheel shimmy.
Often it is the balance. And that is a very easy fix.
When it’s not, every component should be checked for tolerances, sloppiness, looseness, inappropriate strut inflation, all of it... by the book. Often people replace the shimmy dampener and think that’s a fix because it hides it for a while, when really a component is way out of tolerance, bent, broken, whatever.
Also, has to be said... stop landing on the nosewheel. LOL. No seriously. Hold it off.
If it’s touching down almost immediately after the mains, wrong speed or the yoke isn’t touching your gut at the end of the landing.
With practice you can fly it down before it plops down. That engine is heavy up there... help it out with more elevator. All of the elevator.