Let $F$ be a Bianchi newform of level $\mathcal{N}$. When $\mathcal{N}=(N)$ is principal, $F$ is an eigenform for the **Fricke involution** $W_N$ defined by the matrix $\left(\begin{matrix}0&-1\\N&0\end{matrix}\right)$.

The **sign** of $F$ is *minus* the eigenvalue of $W_N$ on $F$. This is equal to the sign of the functional equation satisfied by the L-function attached to $F$. It follows that the analytic rank of $F$ is even when $F$ has sign $+1$ and odd otherwise.

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- Review status: reviewed
- Last edited by John Cremona on 2020-11-02 04:11:17

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- 2020-11-02 04:11:17 by John Cremona (Reviewed)
- 2018-12-13 06:06:19 by Andrew Sutherland (Reviewed)

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