if I have a finite dimensional complex vector space $V$ with a lattice $\Gamma$ in $V$, then I consider the complex linear span in $V\oplus \overline{V}$ of the elements $\gamma \oplus (-\bar{\gamma}$) with $\gamma \in \Gamma$.
How can one describe this span? Is the result the whole $V\oplus \overline{V}$ or smaller?
Remark: a lattice is a full lattice, i.e. free subgroup generated by a real basis of $V$.
Thanks!