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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