Let $\pi_p$ be an irreducible representation of $GL_2(\mathbb{Q}_p)$. Assume $\pi_p$ is ramified,hence it will have a positive conductor. Consider $sym^3(\pi_p)$ which is a representation of $GL_4(\mathbb{Q}_p)$. I want to know how to calculate the conductor of $sym^3(\pi_p)$. What is the relation between conductor of $\pi_p$ and conductor of $sym^3(\pi_p)$? Is there a way to calculate it? If so, then please suggest some good references.
Thank you in advance.
The PhD thesis of Manami Roy (Univ. Oklahoma, 2019) gives a very detailed formula for the conductor of $\operatorname{Sym}^3(\pi_p)$, where $\pi_p$ is the representation of $\mathrm{GL}_2(\mathbf{Q}_p)$ coming from an elliptic curve. See Chapter 5 in particular.
(More precisely, the $\operatorname{Sym}^3$ lifting from $\mathrm{GL}_2$ to $\mathrm{GL}_4$ factors through $\operatorname{GSp}_4$, and Roy computes the conductors of the resulting representations of $\operatorname{GSp}_4(\mathbf{Q}_p)$. But the lifting from $\operatorname{GSp}_4$-representations to $\mathrm{GL}_4$ preserves conductors.)
