Newform ($f=q+{\sum }_{n\ge 2}{a}_n{q}^n$)

A newform is a normalized Hecke eigenform in the new subspace of cusp forms. Its weight, level and Dirichlet character are part of its specification. For a newform, its Fourier coefficients equal its eigenvalues of Hecke operators.

For a newform of weight $k\ge 2$, level $M$ and trivial Dirichlet character, the bad-prime Hecke operator $U_p$ has the displayed possibilities. The local trace relation on the new subspace of cusp forms gives $U_p=-p^{k/2-1}{W}_p$ when $p\parallel M$, where the normalized local Atkin-Lehner operator squares to one; if $p^2\mid M$, the new-vector relation gives $U_p=0$. In particular $|a_p|<p^{k/2}$, a strict inequality useful for proving irreducibility of old blocks.

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 Stein's statement of the theorem.