Symplectomorphism and Hodge decomposition

Question

Let $X$ be some holomorphic symplectic manifold. This means we have choosen some $\omega\in H^{2,0}(X).$ If $f:X\rightarrow X$ is a symplectomorphism, then we have by definition $$f^*\omega=\omega.$$ A symplectomorphism $f^*$ has an eigenvalue $1$ on $H^{2,0}(X)$. However, as was pointed out by Michael Albanese in the comments, this is not a sufficient condition, even if we assume that $X$ has a symplectic form. My question is if there exists a cohomological criterion for an endomorphism to be symplectic for some symplectic form.