Number of non-zero cubic sums

The exponential sums $S_q(a,m)=\sum_{l=1}^q\exp(2\pi i(al^3+ml)q^{-1})$ are considered. For every natural $q$, the explicit formulas for the number of non-zero sums among $S_q(a,0),\dots,S_q(a,q-1)$ are found.