Nilpotence modulo powers of 3, 5 and 7 is proved for Hecke operators on the space of certain modular forms, and is applied to the arithemtic of quadratic forms and Fourier coefficients of modular forms.
Hyunsuk Moon. Yuichiro Taguchi. "$\ell$-adic properties of certain modular forms." Proc. Japan Acad. Ser. A Math. Sci. 82 (7) 83 - 86, March 2006. https://doi.org/10.3792/pjaa.82.83