Twisting of paramodular vectors

Let $F$ be a non-archimedean local field of characteristic zero, let $(π,V)$ be an irreducible, admissible representation of $\GSp(4,F)$ with trivial central character, and let $χ$ be a quadratic character of $F^\times$ with conductor $c(χ)>1$. We define a twisting operator $T_χ$ from paramodular vectors for $π$ of level $n$ to paramodular vectors for $χ\otimes π$ of level $\max(n+2c(χ),4c(χ))$, and prove that this operator has properties analogous to the well-known $\GL(2)$ twisting operator.

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