  Let $S$ be the polynomial ring over a field $K$ in a finite set of variables,
and let $ \mathfrak{m}$ be the graded maximal ideal of $S$. It is known that
for a finitely generated graded $S$-module $M$ and all integers $k\gg 0$, the
module $ \mathfrak{m}^kM$ is componentwise linear. For large $k$ we describe
the pattern of the Betti table of $ \mathfrak{m}^kM$ when $\mathrm{char}(K)=0$
and $M$ is a submodule of a finitely generated graded free $S$-module.
Moreover, we show that for any $k\gg 0$, $ \mathfrak{m}^kI$ has linear
quotients if $I$ is a monomial ideal.
