In recent years, efforts have been made to improve the scope of modeling the dependence of well-known copulas by modifying their mathematical structure. This was the case, among others, of the so-called Celebioglu-Cuadras copula. In this article, we make contributions to this subject by (i) significantly improving an existing result from the literature on the admissible values for a modified version of the Celebioglu-Cuadras copula and (ii) studying a generalization of this modified copula using an additional setting shape parameter. The characteristics of the introduced copulas are discussed, including the shapes of the copula-related functions, various symmetry and dependence structure types, copula inequalities, diverse correlation measures, and bivariate distribution generation. In particular, we highlight the fact that they are ideal for modeling a wide variety of negative-type dependencies and offer an interesting alternative to the Celebioglu-Cuadras and Gumbel-Barnett copulas. Several graphics are produced, and digital work is carried out as support.