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I want to ask the same question with Does the modular form associated to cubic twist of a elliptic curve $E$ corresponds to some twist of $f_E$?

Let $E$ be an elliptic curve defined over $\Bbb Q$ and $f_E$ be the modular form associated with the elliptic curve $E$. Suppose the elliptic curve $E^D$ is a quadratic twist of $E$. I understand that $f_{E^D}$ is a twist of $f_E$ by quadratic character $\mod{D}$. I would like to know when $E^d$ is a cubic twist of E, how are $f_{E^d}$, $f_E$ related?

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  • $\begingroup$ If $E$ has cubic twists, then it has CM by $\mathbb Q(\zeta_3)$. Let $\eta$ be the Hecke character of $\mathbb A_{\mathbb Q(\zeta_3)}^*$ associated to $E$. Then $f$ is the automorphic induction (i.e. theta lift) of $\eta$. Now, let $\chi\colon G_{\mathbb Q(\zeta_3)}\to\mathbb C^*$ be the character corresponding to the Galois extension $\mathbb Q(\sqrt[3]d, \zeta_3)/\mathbb Q(\zeta_3)$, and abusing notation, let $\chi$ also denote the corresponding Hecke character of $\mathbb A_{\mathbb Q(\zeta_3)}^*$. Then $f$ is the automorphic induction of the Hecke character $\eta\otimes\chi$. $\endgroup$
    – Ariel Weiss
    Commented Jul 28, 2023 at 9:24
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    $\begingroup$ This question should be closed as it is precisely a copy of the one linked. It is the prime example of a duplicate. Ariel's comment would fit on the other page nicely, too. $\endgroup$
    – Chris Wuthrich
    Commented Jul 28, 2023 at 10:17

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