The author introduces three functions of two complex variables defined by \[ c_j(z_1,z_2)=\sum_{k=0}^3 \rho^{jk} \exp\{2\pi i(\rho^kz_1+\rho^{2k}z_2)\}\qquad (i=1,2,3), \] where \(\rho=\exp(2\pi i/3)\) and determines their groups of periods. This allows him to give a new form for a product \(\Phi\), related to cubic characters, defined in a previous paper [Math. Notes 7, 284--288 (1970); translation from Mat. Zametki 7, 469--476 (1970; Zbl 0194.35007)]. A new algorithm is proposed to check the conjecture that \(\text{Im}\,\Phi>0\). The author concludes with a remark, that some new results concerning this conjecture were obtained by \textit{J. H. Loxton}. Reviewer's remark: Loxton's paper already appeared in [J. Reine Angew. Math. 268/269, 53--67 (1974; Zbl 0293.10019)].