S_4^12
The following table gives coefficients for the 
1
Miyawaki lift.

The format is: 
(   d, g,m)  coefficient1 coefficient2  [matrix             ]  (true det)

where 
the matrix is given in form [m11 m22 m33 m44 m12 m13 m23 m14 m24 m34] 
d is the determinant of the associated PRIMITIVE form if the matrix is not primitive. 
g is the genus of the associated primitive form (from Gordon Nipp's Table of Quaternary Forms) 
m is the content of the matrix. 
If not primitive, "true det" gives the determinant. 

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(   d, g,m)     coefficient   [matrix             ]  (true det)
(   4, 1,1)             2     [2 2 2 2 0 0 0 1 1 1]
(   5, 1,1)             -5    [2 2 2 2 1 0 0 1 0 1]
(   8, 1,1)             -44   [2 2 2 2 0 0 0 1 1 0]
(   9, 1,1)           -78     [2 2 2 2 1 0 0 0 0 1]
(  12, 1,1)             744   [2 2 2 4 1 1 0 1 0 0]
(  12, 2,1)             -816  [2 2 2 2 0 0 0 1 0 0]
(  13, 1,1)            -377   [2 2 2 4 1 1 0 0 1 0]
(  17, 1,1)            646    [2 2 2 4 1 0 0 1 0 1]
(  20, 2,1)          4080     [2 2 2 4 0 0 0 1 1 1]
(  25, 1,1)          24700    [2 2 4 4 1 1 0 1 1 2]
(  16, 1,1)           -6400   [2 2 2 2 0 0 0 0 0 0]