  We consider for the first time level 7 modular invariant flavour models where
the lepton mixing originates from the breaking of modular symmetry and
couplings responsible for lepton masses are modular forms. The latter are
decomposed into irreducible multiplets of the finite modular group $\Gamma_7$,
which is isomorphic to $PSL(2,Z_{7})$, the projective special linear group of
two dimensional matrices over the finite Galois field of seven elements,
containing 168 elements, sometimes written as $PSL_2(7)$ or $\Sigma(168)$. At
weight 2, there are 26 linearly independent modular forms, organised into a
triplet, a septet and two octets of $\Gamma_7$. A full list of modular forms up
to weight 8 are provided. Assuming the absence of flavons, the simplest
modular-invariant models based on $\Gamma_7$ are constructed, in which
neutrinos gain masses via either the Weinberg operator or the type-I seesaw
mechanism, and their predictions compared to experiment.
