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.



conductor formula
The [PhD thesis of Manami Roy](https://hdl.handle.net/11244/321046) (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.)


