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For $q = e^{2\pi iz}$, let $f(z) = \sum_{n = 1}^{\infty} a(n)q^n\in S_k(N,\chi)$ be a cusp form of integral weight $k \geq 2$ and level $N$ with Dirichlet character $\chi\pmod N$. Let $\alpha_p,\beta_p$ be the local Satake parameters of $f$, i.e. \[1-a(p)X+\chi(p)p^{k-1}X^2=(1-\alpha_p p^{\frac{k - 1}{2}} X)(1-\beta_p p^{\frac{k - 1}{2}} X).\] It is known that if $\chi$ is the trivial character then $\beta_p=\bar{\alpha_p}.$

My question is the following : If $\chi$ is not trivial, on what condition on $f$ we have $\beta_p=\bar{\alpha_p}$ ?

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    $\begingroup$ In general, $\alpha_p \beta_p = \chi(p)$, so that if $\alpha_p \neq 0$, we have that $\beta_p = \chi(p) \alpha_p^{-1}$, which is $\chi(p) \overline{\alpha_p}$ by Deligne's proof of the Ramanujan conjecture. So for $\alpha_p \neq 0$ (which is always the case if $\chi(p) \neq 0$), then $\beta_p = \overline{\alpha_p}$ iff $\chi(p) = 1$. $\endgroup$
    – Peter Humphries
    Commented Apr 8, 2017 at 22:23
  • $\begingroup$ @PeterHumphries, Thank you very much for your answer. $\endgroup$
    – Med
    Commented Apr 8, 2017 at 22:28
  • $\begingroup$ @PeterHumphries Do you want to make that an answer? $\endgroup$
    – Kimball
    Commented Apr 9, 2017 at 21:08

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