Atkin-Lehner theory for nonholomorphic Eisenstein series

Question

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!

Answer 1

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.