  Bell inequalities can be studied both as constraints in the space of
probability distributions and as expectation values of multipartite operators.
The latter approach is particularly useful when considering outcomes as
eigenvalues of unitary operators. This brings the possibility of exploiting the
complex structure of the coefficients in the Bell operators. We investigate
this avenue of though in the known case of two outcomes, and find new Bell
inequalities for the cases of three outcomes and $n=3,4,5$ and $6$ parties. We
find their corresponding classical bounds and their maximum violation in the
case of qutrits. We further propose a novel way to generate Bell inequalities
based on a mapping from maximally entangled states to Bell operators and
produce examples for different outcomes and number of parties.
