(Sorry for my poor english...)

Let $N$ be a positive integer and $f=\sum_{n=1}^{\infty} a(n)q^n\in S_{k}(\Gamma_0(N))\cap K[[q]]$ be a cusp form with number field $\mathbb{Q}(\xi_{N})\subset K$ where $\xi_N$ be a $N$-th root of unity. For $\sigma\in Gal(K/\mathbb{Q})$, $f^{\sigma}$ is defined by \begin{equation} f^{\sigma}=\sum_{n=1}^{\infty}\sigma(a(n))q^n. \end{equation} I already know that $f^{\sigma}\in S_{k}(\Gamma_0(N))$.

Q. Do $f$ and $f^{\sigma}$ have the same order at each cusp $s$ of $\Gamma_0(N)$?

(Sorry for my poor english...)

Let $N$ be a positive integer and $f=\sum_{n=1}^{\infty} a(n)q^n\in S_{k}(\Gamma_0(N))\cap K[[q]]$ be a cusp form with number field $\mathbb{Q}(\xi_{N})\subset K$ where $\xi_N$ be a $N$-th root of unity. For $\sigma\in Gal(K/\mathbb{Q})$, $f^{\sigma}$ is defined by \begin{equation} f^{\sigma}=\sum_{n=1}^{\infty}\sigma(a(n))q^n. \end{equation} I already know that $f^{\sigma}\in S_{k}(\Gamma_0(N))$.

Q. Do $f$ and $f^{\sigma}$ have the same order at each cusp $s$ of $\Gamma_0(N)$?

EDIT : Let $K\subset \mathbb{Q}(\xi_N)$. Do $f$ and $f^{\sigma}$ have the same order at each cusp $s$ of $\Gamma_0(N)$?

(Sorry for my poor english...)

Let $N$ be a positive integer and $f=\sum_{n=1}^{\infty} a(n)q^n\in S_{k}(\Gamma_0(N))\cap K[[q]]$ be a cusp form with number field $\mathbb{Q}(\xi_{N})\subset K$ where $\xi_N$ be a $N$-th root of unity. For $\sigma\in Gal(K/\mathbb{Q})$, $f^{\sigma}$ is defined by \begin{equation} f^{\sigma}=\sum_{n=1}^{\infty}\sigma(a(n))q^n. \end{equation} Already I know that $f^{\sigma}\in S_{k}(\Gamma_0(N))$.

Q. Do $f$ and $f^{\sigma}$ have the same order at each cusp $s$ of $\Gamma_0(N)$?

(Sorry for my poor english...)

Let $N$ be a positive integer and $f=\sum_{n=1}^{\infty} a(n)q^n\in S_{k}(\Gamma_0(N))\cap K[[q]]$ be a cusp form with number field $\mathbb{Q}(\xi_{N})\subset K$ where $\xi_N$ be a $N$-th root of unity. For $\sigma\in Gal(K/\mathbb{Q})$, $f^{\sigma}$ is defined by \begin{equation} f^{\sigma}=\sum_{n=1}^{\infty}\sigma(a(n))q^n. \end{equation} I already know that $f^{\sigma}\in S_{k}(\Gamma_0(N))$.

Q. Do $f$ and $f^{\sigma}$ have the same order at each cusp $s$ of $\Gamma_0(N)$?