Let $F_N$ be the field of modular functions of level $N$ and with Fourier coefficients in $\mathbf Q(\zeta_N)$. We have $$F_N=\mathbf Q(j, f^{(r,s)}_N),$$ where the $f^{(r,s)}_N$ are the Fricke functions, that is, the $x$-coordinates of $N$-torsion points on a suitable elliptic curve defined over $\mathbf Q(j)$. We further have $$\operatorname{Gal}(F_N/\mathbf Q(j))\cong \operatorname{GL}_2(\mathbf Z/N\mathbf Z)/\lbrace \pm 1\rbrace.$$ If $\alpha\in\operatorname{GL}_2(\mathbf Z/N\mathbf Z)$ then we can write $\alpha=\alpha_d\gamma$, where $\alpha_d=\begin{pmatrix}1 \\ &d\end{pmatrix}\gamma$, $\det \gamma =1$, and $\det \alpha =d$. The corresponding automorphisms are given by $f^\gamma=f\circ \gamma$, and if $f=\sum_na_nq^{n/N}$ then $f^{\alpha_d}=\sum_na_n^{\sigma_d}q^{n/N}$, where $\sigma_d\colon Q(\zeta_N)\rightarrow Q(\zeta_N)\colon \zeta_N\mapsto\zeta_N^d$.
Let $\tau_0\in \mathfrak H$ be a CM point. Suppose that $f\in F_N$ and that $f(\tau_0)=0$. Let $\alpha$ be a $2$-by-$2$ matrix with integral coefficients such that $$\alpha\tau_0=\tau_0,\qquad (\det \alpha,N)=1.$$ Let $\overline{\alpha}$ be the image of $\alpha$ in $\operatorname{GL}_2(\mathbf Z/N\mathbf Z)$. How to prove directly that $f^{\overline{\alpha}}(\tau_0)=0$?
