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I am currently reading something about nonholomorphic Eisenstein series $E_\mathfrak{a}(z,1/2+it)$ for $\Gamma_0(q)$, where $\mathfrak{a}$ is a cusp (cf. Iwaniec, H. Spectral Methods of Automorphic Forms). For Maass cusp forms there is Atkin-Lehner theory, but I don't know whether there are similar results for $E_\mathfrak{a}(z,1/2+it)$. Especially, whether $E_\mathfrak{a}(z,1/2+it)$ is an eigenfunction for the Atkin-Lehner operators with explicit eigenvalues. (I don't know much about Atkin-Lehner theory, is there a newform theory for nonholomorphic Eisenstein series?) Can anyone give me a reference about this (even for $q$ being a prime or squarefree number)? Thanks a lot!

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    $\begingroup$ For $GL_2$, all Eisenstein series generate principal series (possibly ramified at some places) everywhere locally. So any questions are not only local, but should be explicitly answerable, since there is no supercuspidal stuff interfering. $\endgroup$ Apr 20, 2015 at 18:10

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I don't believe such a theory exists anywhere in the literature. To the best of my knowledge, there are a couple of results that deal closely with what you are asking.

There exists a theory of newforms for Eisenstein series of the form \[E_{\chi_1,\chi_2}(z,k,\varepsilon) = \sum_{n = 0}^{\infty} a_n e(nz),\] where $z \in \mathbb{H}$, $k$ is a positive integer, $\chi_1,\chi_2, \varepsilon$ are Dirichlet characters for which $\chi_1 \chi_2 = \varepsilon$ and $\epsilon(-1) = (-1)^k$, and the Fourier coefficients are \[a_0 = -\frac{L(0,\chi_1)}{L(1-k,\chi_2)}, \qquad a_n = \sum_{d \mid n} \chi_1\left(\frac{n}{d}\right) \chi_2(d) d^{k-1}.\] This is from Weisenger's thesis, which is available here (see this MO answer).

There is also a theory of the Fourier coefficients, Hecke eigenvalues, and action of the Fricke involution of Eisenstein series of the form \[E_{\mathfrak{a}}(z,\chi,s) = \sum_{\gamma \in \Gamma_{\mathfrak{a}} \backslash \Gamma_0(q)} \overline{\chi}(\gamma) j_{\sigma_{\mathfrak{a}}^{-1} \gamma}(z)^{-k} \Im(\sigma_{\mathfrak{a}}^{-1} \gamma)^s,\] where $\chi$ is a primitive Dirichlet character modulo $q$, $k \in \{0,1\}$ is such that $\chi(-1) = (-1)^k$, $\sigma_{\mathfrak{a}}$ is the scaling matrix for the cusp $\mathfrak{a}$ of $\Gamma_0(q) \backslash \mathbb{H}$ that is singular with respect to $\chi$, and the $j$-factor is such that \[j_{\gamma}(z) = \frac{cz + d}{|cz + d|}, \qquad \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\mathbb{R}).\] The canonical reference for this is probably this paper of Duke, Friedlander, and Iwaniec.


EDIT: Matt Young has recently uploaded a paper to the arXiv where he looks at Eisenstein newforms $E_{\chi_1,\chi_2}(z,s)$, whose Hecke eigenvalues are \[\sum_{ab = n} \chi_1(a) a^{it} \chi_2(b) b^{-it}\] when $s = 1/2 + it$. These, to me, are the most "natural" Eisenstein series from the point of view of Atkin-Lehner theory; Young discusses this aspect in section 9 of his paper.

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