Restriction of translation on an abelian variety to a subvariety

Question

Let $A/\mathbb{C}$ be an abelian variety, and let $X\subset A$ be a closed integral subvariety. Assume that $a\in A$ is such that $T_a(X)\subset X$, where $T_a$ is the translation on $A$ by $a$. Does the restriction $T_a:X\to X$ induce the identity on $H^0(X,\Omega_X^1)$ by pullback global forms on $X$?