Weight 1 for Levels N=2p with Primes p≥5

Abstract

For primes p≥5 there are exactly four new holomorphic eta products of weight 1 for the Fricke group Γ^∗(2p), namely,

$$\left[\frac{2^2, \, p^2}{1, 2p}\right] , \qquad \left[\frac{1^2,(2p)^2}{2, \, p}\right] , \qquad \left[ 2, \, p \right] , \qquad \left[ 1, \, 2p \right] .$$

By Theorem 8.1, each of them is a product of two simple theta series. All of them have denominator 8 if p≡1 mod 3, while for p≡−1 mod 3 the denominators are 8 for the first and second, and 24 for the remaining two eta products. Some of the identities in this subsection are mentioned in (Kahl and Köhler in J. Math. Anal. Appl. 232:312–331, 1999). We begin with the discussion of the case p=5, where we will meet theta series on the fields with discriminants 40, −40 and −4.