Let $F$ be the $\mathbb{Z}$ -module $\mathbb{Z}^{3}$ and let $N$ be the submodule generated by {$(4,-4,4),(-4,4,8),(16,20,40)$}. Find a basis $\left\{ f_{1},f_{2},f_{3}\right\}$ for $F\textrm{ and integers }d_{1},d_{2},d_{3}\textrm{ such that }d_{1}\mid d_{2}\mid d_{3}\textrm{ and }\left\{ d_{1}f_{2},d_{2}f_{2},d_{3}f_{3}\right\}$ is a basis for $N$.
Now I found $d_1 = 4 , d_2 = 12$ and $d_3 = 36$ but I don't know how to find $f_1 , f_2 , f_3$.