Nonlocality is an essential concept that distinguishes quantum from classical models and has been extensively studied in systems of qubits. For higher-dimensional systems, certain results for their two-level counterpart, like Bell violations with stabilizer states and Clifford operators, do not generalize. On the other hand, similar to continuous variable systems, Wigner negativity is necessary for nonlocality in qudit systems. We propose a family of Bell inequalities that inquire correlations related to the Wigner negativity of stabilizer states under the adjoint action of a generalization of the qubit $\pi/8$ gate, which, in the bipartite case, is an abstraction of the CHSH inequality. The classical bound is simple to compute, and a specified stabilizer state maximally violates the inequality among all qudit states based on the Wigner negativity and an inequality between the 1-norm and the maximum norm. The Bell operator not only serves as a measure for the singlet fraction but also quantifies the volume of Wigner negativity. Furthermore, we give deterministic Bell violations, as well as violations with a constant number of measurements, for the Bell state relying on operators innate to higher-dimensional systems than the qudit at hand.