On Atkin and Swinnerton - Dyer Congruence

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 threeterm recursion satisfied by the coefficients of classical Hecke eigen forms. We also show that there is an automorphic L-function over Q whose local factors agree with those of the l-adic Scholl representations attached to the space of noncongruence cusp forms.