Inequality for numbers related to matrices

Question

For an $n \times n$-matrix A let $||A||$ be the maximumsnorm (or another norm if you want). Define the integers $a_{n,m}:= \sup \{ |det(1-A)| | A$ is an $n \times n$ integer matrix with $det(A)=1$ and $||A|| \leq m \}$. Can one give good bounds for $a_{n,m}$ or calculate it?