A note on some diagonal cubic equations over finite fields

Let a prime $p\equiv 1(\text{mod}3)$ and $z$ be non-cubic in ${\mathbb{F}}_p$. Gauss proved that the number of solutions of equation

$$x_1^3+x_2^3+z{x}_3^3=0$$

in ${\mathbb{F}}_p$ was $p^2+\frac{1}{2}(p-1)(9d-c)$, where $c$ was uniquely determined and $d$, except for the sign, was defined by

$$4p=c^2+27d^2,c\equiv 1(\text{mod}3).$$

In 1978, Chowla, Cowles, and Cowles determined the sign of $d$ for the case of 2 being non-cubic in ${\mathbb{F}}_p$. In this paper, we extended the result of Chowla, Cowles and Cowles to finite field ${\mathbb{F}}_q$ with $q=p^k$, $p\equiv 1(\text{mod}3)$, and determined the sign of $d$ for the case of 3 being non-cubic.