# Conductor formula for symmetric square transfer

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?

• You had some $\mathbb\pi$ $\mathbb\pi$ 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 $\mathbb{\pi_p}$ $\mathbb{\pi_p}$. – LSpice Nov 26 '19 at 18:09
• The conductor of the symmetric square should be the conductor of the Rankin-Selberg convolution $\pi \otimes \pi$ minus the conductor of the central character, and I think this should get you the formula you want. – Will Sawin Nov 26 '19 at 18:10