  We prove that there are infinitely many Maass--Hecke cuspforms over the field
$\mathbb{Q}[\sqrt{-3}]$ such that the corresponding symmetric cube $L$-series
does not vanish at the center of the critical strip. This is done by using a
result of Ginzburg, Jiang and Rallis which shows that the symmetric cube
non-vanishing happens if and only if a certain triple product integral
involving the cusp form and the cubic theta function on $\mathbb{Q}[\sqrt{-3}]$
does not vanish. We use spectral theory and the properties of the cubic theta
function to show that the non-vanishing of this triple product occurs for
infinitely many cusp forms. We also formulate a conjecture about the meaning of
the absolute value squared of the triple product which is reminiscent of
Watson's identity.
