Sum rules in the lepton sector provide an extremely valuable tool to classify flavour models in terms of relations between neutrino masses and mixing parameters testable in a plethora of experiments. In this manuscript we identify new leptonic sum rules arising in models with modular symmetries with residual symmetries. These models simultaneously present neutrino mass sum rules, involving masses and Majorana phases, and mixing sum rules, connecting the mixing angles and the Dirac CP-violating phase. The simultaneous appearance of both types of sum rules leads to some non-trivial interplay, for instance, the allowed absolute neutrino mass scale exhibits a dependence on the Dirac CP-violating phase. We derive analytical expressions for these novel sum rules and present their allowed parameter ranges as well as their predictions at upcoming neutrino experiments.
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