The symplectic twistor operator is defined similarly as the twistor operator for Riemannian spin manifolds. It is associated to a symplectic connection and a double cover of the symplectic structure, so called metaplectic structure. The latter was introduced by Kostant in his paper from 1974. If we consider symplectic spinor valued forms, we can define further symplectic twistor operators by decomposing the mentioned forms into invariant subspaces with respect to the double cover of the symplectic group. Analysing the action of the curvature of a chosen symplectic connection, we derive conditions on the curvature under which the symplectic twistor operators form sets of complexes.