2-adic properties of certain modular forms and their applications to arithmetic functions

Non-existence is proved of 2-dimensional mod 2 Galois representations
of the rational number field with small conductor N.
As an application, it is shown that the Hecke action on the space of
modular forms of level 2N is nilpotent in characteristic 2.
As an application, congruences for Fourier coefficients of some modular forms
and some combinatorial quantities are proved.