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 $a_0(\gamma)$ the constant term of the Fourier expansion of $E|_k\gamma$. Here are my questions.

Is it true that for some number field $L$ containing $K$ and the $N$-th roots of unity we have $a_0(\gamma)\in L$ 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 $\mathcal{R}$ denotes the ring of $S$-integers in $L$?

Let $\lambda$ be a prime 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 $\overline{\mathbf{F}}_l$ is cuspidal and let $\gamma\in\mathrm{SL}(2,\mathbf{Z})$. Assume moreover that $a_0(\gamma)$ is $\mathcal{L}$-integral for all $\mathcal{L}\mid\ell$. Is it true that $\ell$ divides the numerator of the norm of $a_0(\gamma)$?

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