Exact holomorphic differentials on a quotient of the Ree curve

We produce several families of exact holomorphic differentials on a quotient X of the Ree curve in characteristic 3, defined by X:yq−y=xq0(xq−x)/Fq (where q0=3s, s≥1 and q=3q02). We conjecture that they span the whole space of exact holomorphic differentials, and prove this in the cases s=1 and s=2, by calculating the kernel of the Cartier operator.