Newforms are eigenforms for the diamond operators

Question

Let $S_k(\Gamma_1(N))^{\textrm{new}}$ denote the new subspace of the space of weight $k$ cusp forms on $\Gamma_1(N)$. Section 5.8 of Diamond and Shurman's modular forms book is supposed to show that if $f \in S_k(\Gamma_1(N))^{\textrm{new}}$, then $f$ is an eigenform for all of the Hecke operators $T_n$ and diamond operators $\left<n\right>$. It seems to me there is a gap though, as the first paragraph of the section ends by noting that it is clear that $f$ is an eigenform for $\left<n\right>$ if $\gcd(n,N) > 1$. since in this case $\left<n\right>$ is the zero operator. Then it says, "Thus the only operators in question are the $T_n$." Well, no... The operators $\left<n\right>$ for $\gcd(n,N) = 1$ are never addressed. In fact it seems to me that then the entire new subspace must be contained in a single $\chi$-eigenspace $S_k(N,\chi)$ for some Dirichlet character $\chi$ modulo $N$, since these eigenspaces are linearly disjoint.