Character sums on an oriented singer conic and explicit Ramanujan double covers

Let $q$ be odd. The trace conic in $\mathbb{F}_{q^3}$ determines a Singer difference set in $\mathbb{F}_{q^3}^\times / \mathbb{F}_q^\times$ and a natural square-class lift to $\mathbb{F}_{q^3}^\times / \mathbb{F}_q^{\times 2}$. We study the odd multiplicative Fourier coefficients of this lift and pr