Equalities of the Petersson inner product for two related modular forms

Question

Let $\Gamma_1(N)$ be the usual congruence subgroup of $\text{SL}_2(\mathbb{Z})$ and let $f\in S_k(\Gamma_1(N))_{\text{new}}$ be a normalized primitive form, and write $f=\sum_{n\geq 1}a_nq^n$. Let $w_N$ be the Fricke, or Atkin-Lehner, operator, which is known to preserve the space of newforms. Define the newform $f^*\in S_k(\Gamma_1(N))_{\text{new}}$ by $f^*(z)=\overline{f(-\overline{z})}$. It can be shown that $w_Nf=\eta_f f^*$ for some $\eta_f\in\mathbb{C}$, known as the Atkin-Lehner pseudo-eigenvalue. For proving that this pseudo-eigenvalue satisfies, amongst other interesting properties, $(-1)^k\eta_{f^*}=\overline{\eta_f}$, I write, with the Petersson inner product $$ \langle f^*,w_nf\rangle_{\Gamma_1(N)}=\overline{\eta_f}\langle f^*,f^*\rangle_{\Gamma_1(N)} $$ $$ \langle f^*,w_nf\rangle_{\Gamma_1(N)}=\langle w_n^\dagger f^*,f\rangle_{\Gamma_1(N)}=(-1)^k\eta_{f^*}\langle \underbrace{f}_{=f^{**}},f\rangle_{\Gamma_1(N)}, $$ but now I need these two inner product to be equal, but I don't see how this is true. I first tried to show that $f\cdot \bar{f}=f^*\cdot\bar{f^*}$, but this didn't seem to hold, and then tried to prove by subsitution of variables ($z\mapsto -\bar{z})$ in the Petersson inner product, but I don't see how the fundamental domain is invariant under this transformation, as it's not given by a $\text{SL}_2(\mathbb{Z})$-matrix. What is a way to show that the inner product are in fact equal?