In the modular symmetry approach to neutrino models, the flavour symmetry emerges as a finite subgroup Γ N of the modular symmetry, broken by the vacuum expectation value (VEV) of a modulus field τ . If the VEV of the modulus τ takes some special value, a residual subgroup of Γ N would be preserved. We derive the fixed points τ S = i, τ ST = (−1 + i √ 3)/2, τ T S = (1 + i √ 3)/2, τ T = i∞ in the fundamental domain which are invariant under the modular transformations indicated. We then generalise these fixed points to τ f = γτ S , γτ ST , γτ T S and γτ T in the upper half complex plane, and show that it is sufficient to consider γ ∈ Γ N . Focussing on level N = 4, corresponding to the flavour group S 4 , we consider all the resulting triplet modular forms at these fixed points up to weight 6. We then apply the results to lepton mixing, with different residual subgroups in the charged lepton sector and each of the right-handed neutrinos sectors. In the minimal case of two right-handed neutrinos, we find three phenomenologically viable cases in which the light neutrino mass matrix only depends on three free parameters, and the lepton mixing takes the trimaximal TM1 pattern for two examples. One of these cases corresponds to a new Littlest Modular Seesaw based on CSD(n) with n = 1 + √ 6 ≈ 3.45, intermediate between CSD (3) and CSD(4). Finally, we generalize the results to examples with three right-handed neutrinos, also considering the level N = 3 case, corresponding to A 4 flavour symmetry. *

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We suggest two alternative schemes to predict lepton mixing angles as well as CP violating phases from a discrete flavor symmetry group combined with CP symmetry. In the first scenario, the flavor and CP symmetry is broken to the residual groups of the structure Z 2 × CP in the neutrino and charged lepton sectors. The resulting lepton mixing matrix depends on two free parameters θ ν and θ l . This type of breaking pattern is extended to the quark sector. In the second scheme, an abelian subgroup contained in the flavor group is preserved by the charged lepton mass matrix and the neutrino mass matrix is invariant under a single remnant CP transformation, all lepton mixing parameter are determined in terms of three free parameters θ 1,2,3 . We derive the most general criterion to determine whether two distinct residual symmetries lead to the same mixing pattern if the redefinition of the free parameters θ ν,l and θ 1,2,3 is taken into account. We have studied the lepton mixing patterns arising from the flavor group S 4 and CP symmetry which are subsequently broken to all of the possible residual symmetries discussed in this work. *

We discussed the scenario that a discrete flavor group combined with CP symmetry is broken to Z 2 × CP in both neutrino and charged lepton sectors. All lepton mixing angles and CP violation phases are predicted to depend on two free parameters θ l and θ ν varying in the range of [0, π). As an example, we comprehensively study the lepton mixing patterns which can be derived from the flavor group ∆(6n 2 ) and CP symmetry. Three kinds of phenomenologically viable lepton mixing matrices are obtained up to row and column permutations. We further extend this approach to the quark sector. The precisely measured quark mixing angles and CP invariant can be accommodated for certain values of the free parameters θ u and θ d . A simultaneous description of quark and lepton flavor mixing structures can be achieved from a common flavor group ∆(6n 2 ) and CP, and accordingly the smallest value of the group index n is n = 7.