  The absolutely maximally entangled (AME) states play key roles in quantum
information processing. We provide an explicit expression of the generalized
Bloch representation of AME states for general dimension $d$ of individual
subsystems and arbitrary number of partite $n$. Based on this analytic formula,
we prove that the trace of the squared support for any given weight is given by
the so-called hyper-geometric function and is irrelevant with the choices of
the subsystems. The optimal point for the existence of AME states is obtained.
