The PhD thesis of Manami Roy (Univ. Oklahoma, 2019) gives a very detailed formula for the conductor of $Sym^3(\pi_p)$, where $\pi_p$ is the representation of $GL_2(Q_p)$ coming from an elliptic curve. See Chapter 5 in particular.

(More precisely, the $sym^3$ lifting from $GL_2$ to $GL_4$ factors through $GSp_4$, and Roy computes the conductors of the resulting representations of $GSp_4(Q_p)$. But lifting from $GSp_4$-representations to $GL_4$ preserves the conductor, if I remember correctly.)

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.)