Exceptional congruences for the coefficients of certain eta-product newforms Original Research Article

Let F(z)=∑n=1∞a(n)qn denote the unique weight 16 normalized cuspidal eigenform on image. In the early 1970s, Serre and Swinnerton-Dyer conjectured thatimagea(p)2p−15≡0,1,2,4 (mod 59),when p≠59 is prime. This was proved in 1983 by Haberland. We describe a general method for proving congruences for the coefficients of eigenforms which arise from odd octahedral complex two-dimensional Galois representations, of which this congruence is the prototypical example. In particular, we prove all such congruences for the coefficients of eta-product newforms.