We continue the study of strong, weak, and dc-weak eigenforms introduced by
Chen, Kiming, and Wiese. We completely determine all systems of Hecke
eigenvalues of level 1 modulo 128, showing there are finitely many. This
extends results of Hatada and can be considered as evidence for the more
general conjecture formulated by the author together with Kiming and Wiese on
finiteness of systems of Hecke eigenvalues modulo prime powers at any fixed
level. We also discuss the finiteness of systems of Hecke eigenvalues of level
1 modulo 9, reducing the question to the finiteness of a single eigenvalue.
Furthermore, we answer the question of comparing weak and dc-weak eigenforms
and provide the first known examples of non-weak dc-weak eigenforms.Comment: 28 pages; Minor revisio