Weight 1 for Fricke Groups Γ^∗(p ² q ²)

Abstract

For primes p>q≥3 there are only two new holomorphic eta products of weight 1 and level p ² q ², namely, η(q ² z)η(p ² z) and η(z)η(q ² p ² z). They belong to the Fricke group, and their orders at ∞ do not allow the construction of eigenforms. Thus our inspection of eta products in this section is confined to the Fricke groups of levels 4p ² for odd primes p. Table 29.1 displays the numbers of new holomorphic eta products also for the groups Γ₀(4p ²). For p≥7 the numbers for Γ^∗(4p ²) are independent from p, and each of these nine eta products is a product of two simple theta series of weight $\frac{1}{2}$ from Theorem 8.1.