A result due to B. Conrad (http://math.stanford.edu/~conrad/papers/prasanna-inv.pdf, Theorem A.1) states that the Atkin-Lehner operator $w_{Q,k}$ is $\mathbb{Z}[1/Q]$-integral on $M_k(\Gamma_0(N))$. In other words, if $f\in M_k(\Gamma_0(N))$ has coefficients in $\mathbb{Z}[1/\ell]$ then $w_{Q,k}(f)$ has coefficients in $\mathbb{Z}[1/\ell]$.

Does anyone know of a corresponding result over $\Gamma_1(N)$? My guess is that $w_{k,Q}$ is $\mathbb{Z}[1/Q][\zeta_Q]$-integral on $M_k(\Gamma_1(N))$ but I can't find a reference for this.

I am however, aware of a weaker result (Theorem 5.4 in https://arxiv.org/abs/1807.00391), which implies that $w_{k,Q}$ is $\mathbb{Q}(\zeta_Q)$-integral on $M_k(\Gamma_1(N))$.

A result due to B. Conrad (http://math.stanford.edu/~conrad/papers/prasanna-inv.pdf, Theorem A.1) states that the Atkin-Lehner operator $w_{Q,k}$ is $\mathbb{Z}[1/Q]$-integral on $M_k(\Gamma_0(N))$. In other words, if $f\in M_k(\Gamma_0(N))$ has coefficients in $\mathbb{Z}[1/Q]$ then $w_{Q,k}(f)$ has coefficients in $\mathbb{Z}[1/Q]$.

Does anyone know of a corresponding result over $\Gamma_1(N)$? My guess is that $w_{k,Q}$ is $\mathbb{Z}[1/Q][\zeta_Q]$-integral on $M_k(\Gamma_1(N))$ but I can't find a reference for this.

I am however, aware of a weaker result (Theorem 5.4 in https://arxiv.org/abs/1807.00391), which implies that $w_{k,Q}$ is $\mathbb{Q}(\zeta_Q)$-integral on $M_k(\Gamma_1(N))$.