Levels N=p ² with Primes p≥3

Abstract

For primes p≥5 there are exactly 6 holomorphic eta products of weight 1 and level N=p ². The only new one among them is η(z)η(p ² z). Since its order at ∞ is $\frac{1 + p^{2}}{24} > 1$, there is little chance to find complementary eta products for the construction of eigenforms which might be represented by Hecke theta series,—at least when we stick to level p ². The chances are improved when we consider η(z)η(p ² z) as an old eta product of level 2p ², and indeed the function η(z)η(25z) will play its rôle in Sect. 20.3.