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Let $F$ be a real quadratic field, and let $f$ be a Hilbert modular form over $F$ of parallel weight 2.

It's known, by a theorem of Rohrlich, that there exist infinitely many Hecke characters of $F$ such that $L(f, \chi, 1) \ne 0$. Does there always exist a Hecke character factoring through the norm map to $\mathbf{Q}$ such that this holds?

(There are some statistics due to Ryan et al, https://math.dartmouth.edu/~jvoight/articles/halfhilbert-022218.pdf, which suggest that this should be true for "almost all" quadratic twists coming from $\mathbf{Q}$; but just one character -- not necessarily quadratic -- would be enough for my purposes!)

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    $\begingroup$ I'm sure you already know this; such a character must be a genus character associated to a pair of primitive quadratic Dirichlet characters $\chi_1$, $\chi_2$ modulo fundamental discriminants $D_1, D_2$ such that $D_1 D_2$ is equal to the fundamental discriminant $D$ of $F$. $\endgroup$
    – Peter Humphries
    Commented May 29, 2020 at 19:34
  • $\begingroup$ No, I don't already know that. Why is it true? $\endgroup$
    – David Loeffler
    Commented May 29, 2020 at 19:48
  • $\begingroup$ (This assertion, if true, would imply that there are only finitely many possible $\chi$. This would contradict the asymptotics of Ryan et al, so I'm a bit sceptical.) $\endgroup$
    – David Loeffler
    Commented May 29, 2020 at 20:02
  • $\begingroup$ Apologies, this is only the case when you are talking about class group characters. In general, the picture is as follows. Let $\omega : F^{\times} \backslash \mathbb{A}_F^{\times} \to \mathbb{C}^{\times}$ be a Hecke character, and suppose that this factors through the norm map $\mathrm{N}_{F/\mathbb{Q}}$: $\omega = \chi \circ \mathrm{N}_{F/\mathbb{Q}}$ for some Hecke character $\chi : \mathbb{Q}^{\times} \backslash \mathbb{A}_{\mathbb{Q}}^{\times} \to \mathbb{C}^{\times}$... $\endgroup$
    – Peter Humphries
    Commented May 29, 2020 at 20:23
  • $\begingroup$ ... Then the automorphic induction of $\omega$ is the Eisenstein series $\chi \boxplus \chi \chi_{F/\mathbb{Q}}$, where $\chi_{F/\mathbb{Q}} : \mathbb{Q}^{\times} \backslash \mathbb{A}_{\mathbb{Q}}^{\times} \to \mathbb{C}^{\times}$ is the quadratic Hecke character associated to $F$. $\endgroup$
    – Peter Humphries
    Commented May 29, 2020 at 20:25

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