Period matrix of abelian surface

Question

Let's construct a complex torus as $(\mathbb{C}^\times)^2/\mathbb{Z}^2$, where the $\mathbb{Z}^2$-action is generated by $$ (z_1,z_2)\mapsto(az_1,bz_2), \ \ \ (z_1,z_2)\mapsto(cz_1,dz_2). $$ My question is, what is the period matrix of this complex torus? I initially thought it is given by $\left[{\begin{array}{cc} \log a & \log b \\ \log c & \log d \ \end{array} } \right]$, but this is in general not symmetric. Any help will be greatly appreciated.