The prime level N=3

Abstract

For Γ₀(3) and weight k=1 there are three holomorphic eta products,

$$[1, 3] , \hspace{3mm}\left[1^3, 3^{- 1}\right] , \hspace{3mm}\left[1^{- 1}, 3^3\right] .$$

The first one is cuspidal and belongs to the Fricke group Γ^∗(3), the others are non-cuspidal. Here we have an illustration for Theorem 3.9 (3): The lattice points on the boundary of the simplex S(2,1) do not belong to S(3,1), and two of the interior lattice points in S(2,1) are on the boundary of S(3,1). At this point it becomes clear that η(z)η(pz) is the only holomorphic eta product of level p and weight 1 for primes p≥5. The eta product η(z)η(3z) is identified with a Hecke theta series for $\mathbb{Q}(\sqrt{- 3})$; the result (11.2) is known from (Dummit et al. in Finite Groups—Coming of Age. Contemp. Math. 45, 89–98, 1985), (Köhler in Math. Z. 197, 69–96, 1988).