Abstract
Following the twisting rule from Ramond Lie superalgebra to Neveu-Schwarz Lie superalgebra, we introduce the twisted version T(p) of a class of Z-graded Lie conformal superalgebras, where p is a nonzero parameter. We completely classify their small rank conformal modules and finite irreducible conformal modules, which turn out to be much richer than those over the original version. As applications, we also develop the representation theory of many new finite Lie conformal superalgebras.
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