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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.

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.

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.

The commuting normal operators $T_{\ell }$, together with the finite-order diamond operators, admit simultaneous diagonalization for the Petersson inner product. Their restrictions to the new subspace of cusp forms have one-dimensional common eigenspaces by the newform multiplicity-one theorem. This additional theorem is the essential newform input: normality alone would allow higher multiplicities. Since every $U_p$ commutes with the good-prime operators and preserves the new space, it preserves each such line and is scalar on it. Normalize its nonzero first Fourier coefficient to one to obtain a newform. This gives the requested simultaneous eigenbasis of the new space.

By definition, every old form is a linear combination of lower-level oldforms by argument dilation. Repeatedly split each lower-level cusp space into its old space and its Petersson inner product orthogonal new space; induction on the level shows that the specified dilations of newforms span the full space. To prove independence, separate distinct primitive eigenforms by their good-prime eigenvalues, using the newform multiplicity-one theorem. For a fixed normalized form, its dilations $f(tz)$ are independent: order the distinct positive integers $t$ increasingly, and the coefficient of the least occurring $q^t$ kills its coefficient in any vanishing linear combination. Repeat for the next least $t$. Thus these dilations are indeed a basis.

For trivial character, part (a) shows that $W_N$ commutes with all the good-prime operators, so it preserves the one-dimensional primitive line. Also $W_N^2=(-1{)}^k$ for the determinant-normalized slash operator; a nonzero trivial-character form has even $k$. ThereforeThe power of $N$ in an unnormalized Fricke eigenvalue depends on the convention. The plain slash action $(cz+d{)}^{-k}f(\gamma z)$ gives $\epsilon N^{-k/2}$, while the Hecke-style action $(\det \gamma {)}^{k-1}(cz+d{)}^{-k}f(\gamma z)$ gives $\epsilon N^{k/2-1}$. The PDF prints $\pm N^{k-1}$; that value requires the deliberate rescaling $w_N=N^{k-1}{W}_N$ and does not follow under either usual convention. Under that rescaling its displayed eigenvalue is $\epsilon N^{k-1}$; all the preceding eigenspace and conjugation statements are unchanged.

If $p$ divides the primitive level $M$ of a trivial-character newform and $N=M{p}^r$, put $e_j=p^{jk/2}f(p^{j}z)$. Then $U_p{e}_0=a_p{e}_0$ and $U_p{e}_j=p^{k/2}{e}_{j-1}$ for $j\ge 1$. The normalized Fricke involution acts by $e_j\mapsto \epsilon e_{r-j}$, where $W_{M}f=\epsilon f$. These two operators act irreducibly: a nonzero invariant subspace contains an eigenvector of $U_p$; its eigenvalue is $a_p$ or zero. Either it contains $e_0$ immediately or applying the reversal and a power of $U_p$ produces $e_0$, using $a_p^2\ne p^k$. Reversal then produces $e_r$, and its successive images produce every basis vector.