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If $f$ is a newform in $S_k(\Gamma_0(N))$ then $f$ is an eigenfunction for both $U_q$ and $W_N$. But in general $U_q$ and $W_N$ do not commute. You can find an example in Shimura, "Introduction to the arithmetic theory of automorphic functions", Remark 3.59. There he constructs eigenfunctions $f$ for $U_q$ such that $W_N f$ is not an eigenfunction for $U_q$, hence $W_N U_q f \neq U_q W_N f$.