  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 $\geq4$,
under some technical assumptions. This requires a "sufficiently rich" Galois
deformation of the symmetric cube which we obtain from the symmetric cube lift
to $\GSp(4)_{/\QQ}$ 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.
