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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**History:**(expand/hide all)

- 2019-03-21 14:19:13 by John Cremona (Reviewed)
- 2019-03-21 14:19:13 by John Cremona
- 2019-03-21 13:27:55 by John Voight
- 2019-03-21 13:27:23 by John Voight
- 2019-03-21 13:26:30 by John Voight
- 2019-03-21 13:25:24 by John Voight
- 2019-03-21 13:25:02 by John Voight

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