Let $G$ be the reductive group $\operatorname{GSp}_{4}$. Let $\pi$ be a smooth admissible cuspidal representation of $\operatorname{GSp}_{4}(\mathbb{A}^{(\infty)})$ of dominant weight. Assume, for caution, that $\pi$ satisfies a multiplicity one hypothesis.

To $\pi$ is attached a representation $\rho$ of the absolute Galois group of $\mathbb{Q}$ unramified outside a finite set of finite places and such that the characteristic polynomial of the Frobenius morphisms $Fr(\ell)$ for $\ell$ outside this set coincides with the Euler factor at $\ell$ of the degree 4 $L$-function of $\pi$. This Galois representation occurs in the degree 3 cohomology of the étale cohomology of a Siegel-Shimura variety.

The image of complex conjugation under $\rho$ is semi-simple so can be chosen to be diagonal with eigenvalues 1 and -1. How many $-1$ are there?

Let $G$ be the reductive group $\operatorname{GSp}_{4}$. Let $\pi$ be a smooth admissible cuspidal representation of $\operatorname{GSp}_{4}(\mathbb{A}^{(\infty)})$ of dominant weight. Assume, for caution, that $\pi$ satisfies a multiplicity one hypothesis.

Fix $p$ an odd prime. To $\pi$ is attached a $p$-adic representation $\rho$ of the absolute Galois group of $\mathbb{Q}$ unramified outside a finite set of finite places and such that the characteristic polynomial of the Frobenius morphisms $Fr\ell$ for $\ell$ outside this set coincides with the Euler factor at $\ell$ of the degree 4 $L$-function of $\pi$. This Galois representation occurs in the degree 3 cohomology of the étale cohomology of a Siegel-Shimura variety.

The image of complex conjugation under $\rho$ is semi-simple so can be chosen to be diagonal with eigenvalues 1 and -1. How many $-1$ are there?