On the (super)cocenter of Cyclotomic Sergeev algebras

We show that cyclotomic Sergeev algebra $\mathfrak{h}_n^g$ is symmetric when the level is odd and supersymmetric when the level is even. We give an integral basis for ${\rm Tr}(\mathfrak{h}_n^g)_{\overline{0}}$, and recover Ruff's result on the rank of ${\rm Z}(\mathfrak{h}_n^g)_{\bar{0}}$ when the level is odd. We obtain a generating set of ${\rm SupTr}(\mathfrak{h}_n^g)_{\overline{0}}$, which gives an upper bound of the dimension of ${\rm Z}(\mathfrak{h}_n^g)_{\bar{0}}$ when the level is even.