Prime-power congruences for level-one eigenforms

We study prime-power congruences for the prime-to-$p$ Hecke eigensystems of normalized cuspidal level-one eigenforms as the weight varies, and we address the problem of classifying such systems for small prime powers. We construct Hecke-equivariant transfer maps between level-one modular-symbol modules via multiplication by suitable lifts of Dickson invariants; these maps reduce the problem to finite computation and propagate Hecke operator relations through all weights. For $p=2,3,5,7$, we obtain all-weight classifications modulo $2^8$, $3^4$, $5^3$, and $7^2$, respectively.