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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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• Last edited by Andrew Sutherland on 2018-12-13 06:06:19
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