I am trying to work through Diamond and Shurman's A First Course in Modular Forms but am stuck in one of the exercises. Exercise 5.11.2 asks us to show that given a normalized eigenform $f\in S_k(\Gamma_1(N))$ with eigenvalues $a_p$ for Hecke operators $T_p$, $f$ has eigenvalue $\overline{a_p }$ under the adjoint $T_p^*$ with respect to the Petersson inner product.
Assuming that $T_p^*f= b_p f$, we see that $a_p||f||^2 = \langle a_pf,f\rangle = \langle T_pf,f\rangle = \langle f,T_p^*f\rangle = \langle f,b_pf\rangle=\overline{b_p}||f||^2$, which gives us $b_p = \overline{a_p}$ as needed. However, I do not see why $f$ has to be an eigenvector for the operators $T_p^*$ for all $p$.
For $p\nmid N$, $T_p^* = \langle p\rangle^{-1}T_p$, and since $f$ is an eigenvector for both $\langle p\rangle$ and $T_p$, so in this case I can see that $f$ is an eigenform. But what about the case $p\mid N$? The given hint just assumes that $T_p^*f= b_p f$ for all $p$ and proceeds as in the last paragraph.
