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A modular form $f$ for $\GL(2)$ over a number field $F$ has complex multiplication (CM) if it is equal to its twist by a nontrivial Hecke character $\chi$ over $F$, or equivalently $a_{\frak{p}} = \chi(\frak{p}) a_{\frak{p}}$ for all good primes $\frak{p} \nmid \frak{N}$. Such a character $\chi$ is necessarily quadratic, and $$a_{\frak{p}} = 0$$ for all good primes $$\frak{p}$$ such that $$\chi(\frak{p}) = -1$$.

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• Review status: reviewed
• Last edited by John Cremona on 2019-03-21 14:19:13
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