= Newform multiplicity-one theorem
On the <new subspace of cusp forms>, a common <eigenspace> for the <Hecke operators> away from the level has dimension one. Distinct <newforms>, even at different primitive levels, have different systems of these <eigenvalues>. Consequently the commuting bad-prime operators also act by scalars on these lines. This is the multiplicity-one part of Atkin-Lehner-Li theory, not a consequence of <normal operator> diagonalization alone. See https://wstein.org/books/modform/modform/newforms.html#atkin-lehner-li-theory[Stein's statement of the theorem].
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