On Greenberg's $L$-invariant of the symmetric sixth power of an ordinary cusp form

We derive a formula for Greenberg’s $L$-invariant of Tate twists of the symmetric sixth power of an ordinary non-CM cuspidal newform of weight $\ge 4$, under some technical assumptions. This requires a “sufficiently rich” Galois deformation of the symmetric cube which we obtain from the symmetric cube lift to $\text{GSp}(4{)}_{\mathrm{/}\text{QQ}}$ of Ramakrishnan–Shahidi and the Hida theory of this group developed by Tilouine–Urban. The $L$-invariant is expressed in terms of derivatives of Frobenius eigenvalues varying in the Hida family. Our result suggests that one could compute Greenberg’s $L$-invariant of all symmetric powers by using appropriate functorial transfers and Hida theory on higher rank groups.