In 1973, Swinnerton-Dyer (Modular functions of one variable III: proceedings International Summer School University of Antwerp, RUCA, 17 July–3 August 1972. Springer, pp 1–55, 1973) classified congruences modulo primes \(\ell \) for the coefficients of normalized eigenforms in weights \(k \in \{12, 16, 18, 20, 22, 26\}\) on \({{\,\textrm{SL}\,}}_2(\mathbb {Z})\) into three types according to the possible images for the associated mod \(\ell \) Galois representations when the images do not contain \({{\,\textrm{SL}\,}}_2(\mathbb {F}_\ell )\) . In this paper, we classify congruences for the coefficients of eta-quotient newforms of two of the three types: those arising from mod \(\ell \) representations whose images are contained in a Borel subgroup of \({{\,\textrm{GL}\,}}_2(\mathbb {F}_\ell )\) and those arising from representations whose images lie in the normalizer of a Cartan subgroup, but not in the Cartan subgroup itself. We also give examples of extensions of these congruences to prime power modulus, as Swinnerton-Dyer did in (Modular functions of one variable III: proceedings International Summer School University of Antwerp, RUCA, 17 July–3 August 1972. Springer, pp 1–55, 1973) for coefficients of the Delta function. In 2003, independently Kiming and Verrill (J Number Theory 110(2):236–266, 2005) and Boylan (J Number Theory 98(2):377–389, 2003) classified congruences for eta-quotient newforms of the third type, namely those arising from mod \(\ell \) representations with projective image \(A_4\) , \(S_4\) , or \(A_5\) . Therefore, our work completes the classification of exceptional congruences, in the sense of Swinnerton-Dyer, for eta-quotient newforms.
