I understand and share quid's bewilderment. There are easily more than 5000 articles
whose main theme can be considered, without stretching too much, as "congruencies between Fourier coefficients of modular forms - and generalizations", including some of the most famous of the last four decades, like the one proving Fermat's last theorem.
The theory of congruences between modular forms has blossomed in many directions.
You may want to look at work of Serre, Katz, Hida, Mazur, Ribet, Wiles (and others) in the 70's and 80's for the beginning of the story.
Edit: There are so many directions of generalizations that it's hard to decide where to begin with. Swinnerton-Dyer works with congruences between two modular forms of the same level and the same weights. Those congruences are in some sense "accidental". Several papers by Hida
(like this one "congruences between cusp forms and special values of their Zeta functions")
hace generalized this line of thought. Besides, one can consider congruences between forms
of the same level and various weights -- and you get the theory of p-adic modular forms, Hida
families, eigencurve, etc. Or congruences between forms of the same weights and different
level -- you get to the level-raising and level-lowering result of Ribet's and other.
Ribet's ICM talk is a good introduction to this. Then you get all the generalization to other automorphic forms...