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A Bianchi newform $F$ of level $\mathcal{N}$ over an imaginary quadratic field $K$ is said to have Complex Multiplication or CM if it is equal to its own twist by a non-trivial Hecke character $\chi$. Such a character $\chi$ is necessarily quadratic. The CM property is characterised in terms of the Hecke eigenvalues $a_{\frak{p}}$of $F$ by the property that \[ a_{\frak{p}} = \chi(\frak{p}) a_{\frak{p}} \] for all primes $\frak{p}$ not dividing the level. In particular, $a_{\frak{p}}=0$ for all primes $\frak{p}$ for which $\chi(\frak{p})=-1$.

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  • Review status: beta
  • Last edited by John Cremona on 2017-07-14 14:12:35
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