Let $f\in S_k(\Gamma_0(N))$ be a cusp form for $N>1$. Consider the following operators acting on $f$ via the natural action of $GL_2^{+}(\mathbb{R})$ :

$$ W_N=\begin{pmatrix} 0 & -1\\ N & 0 \end{pmatrix}$$

$$ U_q=\sum\limits_{i=0}^{q-1}\begin{pmatrix} q & i\\ 0 & q \end{pmatrix}$$ for prime $q\mid N$.

Does $U_qW_Nf=W_NU_qf$?

Remark : The action for $\gamma=\begin{pmatrix} a & b\\ c & d\end{pmatrix}\in GL_2^{+}(\mathbb{R})$ is given by :

$$\gamma f(z)= (\operatorname{det}(\gamma))^{k/2}(cz+d)^{-k} f\left(\dfrac{az+b}{cz+d}\right)$$.

Any help is deeply appreciated.

Let $f\in S_k(\Gamma_0(N))$ be a cusp form for $N>1$. Consider the following operators acting on $f$ via the natural action of $GL_2^{+}(\mathbb{R})$ :

$$ W_N=\begin{pmatrix} 0 & -1\\ N & 0 \end{pmatrix}$$

$$ U_q=\sum\limits_{i=0}^{q-1}\begin{pmatrix} q & i\\ 0 & q \end{pmatrix}$$ for prime $q\mid N$.

Does $U_qW_Nf=W_NU_qf$?

Remark : The action for $\gamma=\begin{pmatrix} a & b\\ c & d\end{pmatrix}\in GL_2^{+}(\mathbb{R})$ is given by :

$$\gamma f(z)= (\operatorname{det}(\gamma))^{k/2}(cz+d)^{-k} f\left(\dfrac{az+b}{cz+d}\right).$$

Any help is deeply appreciated.