Paramodular newvectors and twists

Question

In the book Local Newforms for GSp(4), Roberts and Schmidt have defined a theory of "new vectors" for smooth representations of $GSp_4$ over a nonarchimedean local field $F$ with trivial central character, using the paramodular congruence subgroups.

If $\pi$ is a generic unramified representation of $GSp_4(F)$ with trivial central char, and $\chi$ is a ramified quadratic character of $F^\times$, then $\pi \otimes \chi$ again has trivial central character; so there must be some vector in $\pi$ whose image in $\pi \otimes \chi$ is the new vector of the twist.

Can one write down an explicit element of $\mathbf{C}[GSp_4(F)]$ which sends the spherical vector of $\pi$ to the paramodular new vector of $\pi \otimes \chi$?

For $GL_2$ it's well known that the operator $\sum_{u \in (\mathcal{O}_F / \varpi)^\times} \chi(u)^{-1} \begin{pmatrix} 1 & u/\varpi \\ 0 & 1 \end{pmatrix}$ does the job, so I'm hoping for something like this. There is a paper by Andrianov "Twisting of Siegel modular forms" which seems related, but he doesn't use the paramodular subgroups.

Answer 1

I've stumbled across an answer to this old question of mine so I'm going to answer it myself. The following paper:

Johnson-Leung, Jennifer; Roberts, Brooks, Twisting of paramodular vectors, Int. J. Number Theory 10, No. 4, 1043-1065 (2014). (preprint version, official journal version)

addresses this exact question, giving an explicit (but lengthy) formula for a twisting map from newvectors of $\pi$ to newvectors of $\pi \otimes \chi$ (for any paramodular $\pi$, not necessarily unramified).