Supppose f and g are two normalized classical eigenforms of tame
level N weight characters \kqppa and \tau on \Z_p^*. Suppose
f(q)\con g(q) modulo p^m.
Then, when $p$ is odd, I can show
\kappa\con \tau modulo p^m.
When f and g have level 1 and rational coefficients Serre showed the
same is true when p=2, Thm. 1 of sect. 1.3 of Formes modulaires et
fonctions zeta p-adic, but I don't believe it is true, in general, for
p=2. I guess, I would first look at forms on X_1(4). (I am still
trying to explain your computations of last Fall.)
Robert