  Inspired by the structure of top-down derived models endowed with modular
flavor symmetries, we investigate the yet phenomenologically unexplored binary
dihedral group 2D_3. After building the vector-valued modular forms in the
representations of 2D_3 with small modular weights, we systematically classify
all (Dirac and Majorana) mass textures of fermions with fractional modular
weights and all possible 2+1-family structures. This allows us to explore the
parameter space of fermion models based on 2D_3, aiming at a description of
both quarks and leptons with a minimal number of parameters and best
compatibility with observed data. We consider the separate possibilities of
neutrino masses generated by either a type-I seesaw mechanism or the Weinberg
operator. We identify a model that, besides fitting all known flavor
observables, delivers predictions for six not-yet measured parameters and
favors normal-ordered neutrino masses generated by the Weinberg operator. It
would be interesting to figure out whether it is possible to embed our model
within a top-down scheme, such as T2/Z4 heterotic orbifold compactifications.
