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In the modular symmetry approach to neutrino models, the flavour symmetryemerges as a finite subgroup $\Gamma_N$ of the modular symmetry, broken by thevacuum expectation value (VEV) of a modulus field $\tau$. If the VEV of themodulus $\tau$ takes some special value, a residual subgroup of $\Gamma_N$would be preserved. We derive the fixed points $\tau_S=i$,$\tau_{ST}=(-1+i\sqrt{3})/2$, $\tau_{TS}=(1+i\sqrt{3})/2$, $\tau_T=i\infty$ inthe fundamental domain which are invariant under the modular transformationsindicated. We then generalise these fixed points to $\tau_f=\gamma\tau_S$,$\gamma\tau_{ST}$, $\gamma\tau_{TS}$ and $\gamma\tau_{T}$ in the upper halfcomplex plane, and show that it is sufficient to consider$\gamma\in\Gamma_{N}$. Focussing on level $N=4$, corresponding to the flavourgroup $S_4$, we consider all the resulting triplet modular forms at these fixedpoints up to weight 6. We then apply the results to lepton mixing, withdifferent residual subgroups in the charged lepton sector and each of theright-handed neutrinos sectors. In the minimal case of two right-handedneutrinos, we find three phenomenologically viable cases in which the lightneutrino mass matrix only depends on three free parameters, and the leptonmixing takes the trimaximal TM1 pattern for two examples. One of these casescorresponds to a new Littlest Modular Seesaw based on CSD$(n)$ with$n=1+\sqrt{6}\approx 3.45$, intermediate between CSD$(3)$ and CSD$(4)$.Finally, we generalize the results to examples with three right-handedneutrinos, also considering the level $N=3$ case, corresponding to $A_4$flavour symmetry.

Modular $S_4$ and $A_4$ Symmetries and Their Fixed Points: New Predictive Examples of Lepton Mixing