= Solution
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 $k$. Therefore
$$
\boxed{W_Nf=\varepsilon f,\qquad\varepsilon\in\{1,-1\}.}
$$
The power of $N$ in an unnormalized Fricke eigenvalue depends on the convention. The plain slash action $(cz+d)^{-k}f(\gamma z)$ gives $\varepsilon N^{-k/2}$, while the Hecke-style action $(\det\gamma)^{k-1}(cz+d)^{-k}f(\gamma z)$ gives $\varepsilon N^{k/2-1}$. 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 $\varepsilon N^{k-1}$; all the preceding eigenspace and conjugation statements are unchanged.
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