I have a question about a formula given in complex analysis from Freitag. In Chapter VI, right above lemma 6.5, it talks about the mapping $\left[ \Gamma[2], r/2, vv_{\vartheta}^r \right] \longrightarrow \left[ \Gamma[2], r/2, v^{(N,r)} v_{\vartheta}^r \right]$ given by $f \mapsto f \vert N^{-1}$ for a fixed matrice $N \in \mathrm{SL_2}(\mathbb{Z})$. Right under this, it claims that for the new multiplicator system $v^{(N,r)}$ the formula $v^{(N,r)}(M) = v(NMN^{-1})\frac{v_{\vartheta}(NMN^{-1})}{v_{\vartheta}(M)}$ holds. Now I dont understand where this comes from. In remark 5.7 something similar is given, but there is an additional algebraic sign and the conjugation with N is the other way around. It must come somehow from the fact that $\Gamma[2]$ is a normal subgroup, but I really can't find a way to prove that.
Thanks for answers! Hari.