Congruences modulo prime powers of Hecke eigenvalues in level 1

  • Nadim RustomEmail author


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.


Modular forms Congruences 

Mathematics Subject Classification

Primary 11F33 Secondary 11F80 



The author would like to thank Ian Kiming and Gabor Wiese for many interesting discussions on the topic of eigenforms modulo prime powers over the year, as well as Shaunak Deo and Ming-Lun Hsieh for helpful discussions and comments on this work. The author would also like to thank the anonymous referee for thoroughly reading the manuscript and providing numerous valuable comments and suggestions, in particular suggesting how to fill an earlier gap in the proof of Proposition 10.2. All relevant computations were done using Sage. This research was supported by a Postdoctoral Fellowship at the National Center for Theoretical Sciences, Taipei, Taiwan.


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Authors and Affiliations

  1. 1.Mathematics DepartmentKoç University, Rumelifeneri YoluSarıyerTurkey

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