On Atkin and Swinnerton-Dyer congruence relations (2)

Abstract

In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the p-adic analogue of the three-term recursion satisfied by the coefficients of classical Hecke eigenforms. We also show that there is an automorphic L-function over \({\mathbb{Q}}\) whose local factors agree with those of the l-adic Scholl representations attached to the space of noncongruence cusp forms.