On the average number of octahedral modular forms

λf (p) = Tr(ρ(Frobp)) χ(p) = det(ρ(Frobp)) for all p not dividing N . Following [4], we define SArtin 1/4,k (N,χ) to be the finite set of primitive weight k cupsidal eigenforms which admit an associated Galois representation. If ρ : Gal(Q̄/Q)→ GL2(C) is a Galois representation, we define Pρ to be the composition of ρ with the natural projection GL2(C)→ PGL2(C). Two-dimensional complex Galois representations fall naturally into four types; we call ρ dihedral, tetrahedral, octahedral, or icosahedral according as the projectivized image Pρ(Gal(Q̄/Q)) is isomorphic to a dihedral group, A4, S4, or A5. Cusp forms associated to Galois representations are classified likewise. The latter three types are called “exotic”; it is widely believed that the number of exotic cusp forms of level N is at most N . The first results in this direction are due to Duke [3]. These results were later sharpened by Wong [6] and Michel and Venkatesh [4]. The latter authors proved that