Let $\pi$ be an automorphic cuspidal representation of $\operatorname{GL}_2(\mathbb A_{\mathbb Q})$, and let $\Pi = \operatorname{sym}^2(\mathbb\pi)$, which is a representation of $\operatorname{GL}_3(\mathbb A_{\mathbb Q})$. Let $\pi= \bigotimes'\pi_p$ and $\Pi= \bigotimes'\Pi_p$. Let $c(\Pi_p)$ and $c(\pi_p)$ be the conductors for $\Pi_p$ and $\pi_p$ respectively. Can we get some relation between the conductors $c(\Pi_p)$ and $c(\pi_p)$ using theorem 6.5 of Bushnell - Local Rankin–Selberg convolutions for $\operatorname{GL}_n$: Explicit conductor formula or any other way?
Thank you in advance.
$\mathbb\pi$and some $\pi$$\pi$; since I think the latter is more usual, I edited for uniformity. If you do restore\mathbb, note that it shouldn't apply to everything; so, for example, $\mathbb\pi_p$$\mathbb\pi_p$, not $\mathbb{\pi_p}$$\mathbb{\pi_p}$. $\endgroup$ – LSpice Nov 26 '19 at 18:09