Sign of a Bianchi newform (reviewed)

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.