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!



Atkin-Lehner theory for nonholomorphic Eisenstein series
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](http://benjaminlinowitz.com/Weisinger-Thesis.pdf) (see [this MO answer](https://mathoverflow.net/questions/196242/weisingers-thesis)).


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](http://link.springer.com/article/10.1007%2Fs002220200223) of Duke, Friedlander, and Iwaniec.




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EDIT: Matt Young has recently uploaded [a paper](https://arxiv.org/pdf/1710.03624.pdf) 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.


