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Take a holomorphic cusp newform, say $f \in S_k(N)^\mathrm{new}$, for a squarefree level $N$. It is an eigenvalue of the Atkin-Lehner operator, and this feature allows to express its root number as $$\varepsilon_f = i^k \mu(N) \sqrt{N} \lambda_f(N)$$

Is there any such expression for the root number, only in terms of its level, conductor and coefficients, in the case of Maass forms?

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    $\begingroup$ (1) This formula only holds for squarefree level! Indeed, $\lambda_f(p) = 0$ if $f$ is a newform of level $q \equiv 0 \pmod{p^2}$ and principal nebentypus. (2) Essentially the same formula holds for (even or odd) Maass forms of weight zero and principal nebentypus, without the $i^k$ term; this is essentially shown in Proposition 8.1 of Duke-Friedlander-Iwaniec. $\endgroup$
    – Peter Humphries
    Commented Dec 18, 2017 at 10:59
  • $\begingroup$ @PeterHumphries I corrected (1), thanks for the comment. I seems perfect for (2), I will dig into Duke-Friedlander-Iwaniec. $\endgroup$
    – Desiderius Severus
    Commented Dec 18, 2017 at 11:09
  • $\begingroup$ A caveat on DFI: ostensibly, the result is for primitive nebentypus, but Proposition 8.1 is valid regardless in the imprimitive case. Also there are some minor mistakes in the proof and statement: see the bottom of page 8 of arxiv.org/abs/1710.03624 $\endgroup$
    – Peter Humphries
    Commented Dec 18, 2017 at 11:13

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