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Here is my setting: Let $E\in\mathcal{M}_k(\Gamma_0(N))$ be an Eisenstein series (of trivial Nebentypus) that is a normalized eigenform for all the Hecke operators at level $\Gamma_0(N)$. Assume that the Fourier expansion of $E$ lies in $\mathcal{O}_K[[q]]$ where $\mathcal{O}K$ \mathcal{O}_K$ is the integer ring of some number field $K$. For any $\gamma\in\mathrm{SL}(2,\mathbf{Z})$ let us denote by $a{0,\gamma}(E)$ a_0(\gamma)$ the constant term of the Fourier expansion of $E|k\gamma$. E|_k\gamma$. Here are my questions: \begin{enumerate} \item .

  1. Is that it true that for some number field $L$ containing $K$ and the $N$-th roots of unity we have : $$ a{0,\gamma}(E)\in L\quad\textrm{and}\quad E|k\gamma\in a{0,\gamma}(E)+q^{1/N}\mathcal{O}{L,S}[[q^{1/N}]], $a_0(\gamma)\in L$ and $E|_k\gamma\in a_0(\gamma)+q^{1/N}\mathcal{R}[[q^{1/N}]]$ where $S$ is a finite set of places in $L$ containing the finite places above $N$ and the archimedian ones and $\mathcal{O}{L,S}$ \mathcal{R}$ denotes the ring of $S$-integers in $L$?\item

  2. Let $\lambda$ be a prime ideal in $\mathcal{O}_K$ of residue characteristic $\ell\nmid N$. Since by assumption the Fourier expansion of $E$ lies in $\mathcal{O}K[[q]]$ \mathcal{O}_K[[q]]$, one may reduce $E$ modulo $\lambda$. Assume that the corresponding modular form over $\overline{\mathbf{F}}{\ell}$ \overline{\mathbf{F}}_l$ is cuspidal and let $\gamma\in\mathrm{SL}(2,\mathbf{Z})$. Assume moreover that $a_{0,\gamma}(E)$ a_0(\gamma)$ is $\mathcal{L}$-integral for any all $\mathcal{L}\mid\ell$. Is it true that $\ell$ divides the numerator of the norm of $a_{0,\gamma}(E)$? \end{enumerate} a_0(\gamma)$?

I believe the answer to both questions is ``yes'' yes, but I'd like to have a proof or a reference. Thanks for your help!

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Fourier expansion of Eisenstein series at various cusps

Here is my setting: Let $E\in\mathcal{M}_k(\Gamma_0(N))$ be an Eisenstein series (of trivial Nebentypus) that is a normalized eigenform for all the Hecke operators at level $\Gamma_0(N)$. Assume that the Fourier expansion of $E$ lies in $\mathcal{O}_K[[q]]$ where $\mathcal{O}K$ is the integer ring of some number field $K$. For any $\gamma\in\mathrm{SL}(2,\mathbf{Z})$ let us denote by $a{0,\gamma}(E)$ the constant term of the Fourier expansion of $E|k\gamma$. Here are my questions: \begin{enumerate} \item Is that true that for some number field $L$ containing $K$ and the $N$-th roots of unity we have: $$ a{0,\gamma}(E)\in L\quad\textrm{and}\quad E|k\gamma\in a{0,\gamma}(E)+q^{1/N}\mathcal{O}{L,S}[[q^{1/N}]], $$ where $S$ is a set of places in $L$ containing the finite places above $N$ and the archimedian ones and $\mathcal{O}{L,S}$ denotes the ring of $S$-integers in $L$? \item Let $\lambda$ be a prime ideal in $\mathcal{O}_K$ of residue characteristic $\ell\nmid N$. Since by assumption the Fourier expansion of $E$ lies in $\mathcal{O}K[[q]]$ one may reduce $E$ modulo $\lambda$. Assume that the corresponding modular form over $\overline{\mathbf{F}}{\ell}$ is cuspidal and let $\gamma\in\mathrm{SL}(2,\mathbf{Z})$. Assume moreover that $a_{0,\gamma}(E)$ is $\mathcal{L}$-integral for any $\mathcal{L}\mid\ell$. Is it true that $\ell$ divides the numerator of the norm of $a_{0,\gamma}(E)$? \end{enumerate} I believe the answer to both questions is ``yes'' but I'd like a proof or a reference. Thanks for your help!