#Defining polynomial for the field of definition of the Gram-matrix: mpapmm3 := delta^4-26*delta^3+243*delta^2-970*delta+1397; #Among its real roots the ones in the following postions #(roots in increasing order) #yield the relevant real Gram-matrices: rootapmm3:=[2, 3]; #(Warning: Use at least 10 digits) #Minimal polynomial of trace of Gram-matrix: Trapmm3 := lambda^4-26*lambda^3+243*lambda^2-970*lambda+1397; #Numerical approximations of Eigenvalues of relevant Gram-matrices: EVGapmm32 := [1.50589466677606, 1.50589466677606, 2.2769479397813]; EVGapmm33 := [1.66540420066696, 1.66540420066696, 3.48491471391155]; #Formal Gram-matrix: Gapmm3 := Matrix(12,12,[ [1/12*delta^2-delta+13/4, 1/12*delta^2-delta+11/4, 1/12*delta-1/2, 1/12*delta-1/2, 1/12*delta-1/2, 1/12*delta^2-delta+11/4, -5/12*delta+5/2, -5/12*delta^2+5*delta-59/4, 7/12*delta-7/2, -5/12*delta^2+5*delta-59/4, -5/12*delta+5/2, 7/12*delta^2-7*delta+83/4], [1/12*delta^2-delta+11/4, 1/12*delta^2-delta+13/4, 1/12*delta-1/2, 1/12*delta-1/2, -5/12*delta+5/2, -5/12*delta^2+5*delta-59/4, 1/12*delta-1/2, 1/12*delta^2-delta+11/4, -5/12*delta+5/2, 7/12*delta^2-7*delta+83/4, 7/12*delta-7/2, -5/12*delta^2+5*delta-59/4], [1/12*delta-1/2, 1/12*delta-1/2, -1/12*delta^2+7/6*delta-13/4, 1/12*delta^2-7/6*delta+41/12, 1/12*delta^2-7/6*delta+41/12, 1/12*delta-1/2, -1/12*delta^2+7/6*delta-15/4, -5/12*delta+5/2, 1/12*delta^2-7/6*delta+47/12, -5/12*delta+5/2, -1/12*delta^2+7/6*delta-15/4, 7/12*delta-7/2], [1/12*delta-1/2, 1/12*delta-1/2, 1/12*delta^2-7/6*delta+41/12, -1/12*delta^2+7/6*delta-13/4, -1/12*delta^2+7/6*delta-15/4, -5/12*delta+5/2, 1/12*delta^2-7/6*delta+41/12, 1/12*delta-1/2, -1/12*delta^2+7/6*delta-15/4, 7/12*delta-7/2, 1/12*delta^2-7/6*delta+47/12, -5/12*delta+5/2], [1/12*delta-1/2, -5/12*delta+5/2, 1/12*delta^2-7/6*delta+41/12, -1/12*delta^2+7/6*delta-15/4, -1/12*delta^2+7/6*delta-13/4, 1/12*delta-1/2, 1/12*delta^2-7/6*delta+47/12, 7/12*delta-7/2, -1/12*delta^2+7/6*delta-15/4, 1/12*delta-1/2, 1/12*delta^2-7/6*delta+41/12, -5/12*delta+5/2], [1/12*delta^2-delta+11/4, -5/12*delta^2+5*delta-59/4, 1/12*delta-1/2, -5/12*delta+5/2, 1/12*delta-1/2, 1/12*delta^2-delta+13/4, 7/12*delta-7/2, 7/12*delta^2-7*delta+83/4, -5/12*delta+5/2, 1/12*delta^2-delta+11/4, 1/12*delta-1/2, -5/12*delta^2+5*delta-59/4], [-5/12*delta+5/2, 1/12*delta-1/2, -1/12*delta^2+7/6*delta-15/4, 1/12*delta^2-7/6*delta+41/12, 1/12*delta^2-7/6*delta+47/12, 7/12*delta-7/2, -1/12*delta^2+7/6*delta-13/4, 1/12*delta-1/2, 1/12*delta^2-7/6*delta+41/12, -5/12*delta+5/2, -1/12*delta^2+7/6*delta-15/4, 1/12*delta-1/2], [-5/12*delta^2+5*delta-59/4, 1/12*delta^2-delta+11/4, -5/12*delta+5/2, 1/12*delta-1/2, 7/12*delta-7/2, 7/12*delta^2-7*delta+83/4, 1/12*delta-1/2, 1/12*delta^2-delta+13/4, 1/12*delta-1/2, -5/12*delta^2+5*delta-59/4, -5/12*delta+5/2, 1/12*delta^2-delta+11/4], [7/12*delta-7/2, -5/12*delta+5/2, 1/12*delta^2-7/6*delta+47/12, -1/12*delta^2+7/6*delta-15/4, -1/12*delta^2+7/6*delta-15/4, -5/12*delta+5/2, 1/12*delta^2-7/6*delta+41/12, 1/12*delta-1/2, -1/12*delta^2+7/6*delta-13/4, 1/12*delta-1/2, 1/12*delta^2-7/6*delta+41/12, 1/12*delta-1/2], [-5/12*delta^2+5*delta-59/4, 7/12*delta^2-7*delta+83/4, -5/12*delta+5/2, 7/12*delta-7/2, 1/12*delta-1/2, 1/12*delta^2-delta+11/4, -5/12*delta+5/2, -5/12*delta^2+5*delta-59/4, 1/12*delta-1/2, 1/12*delta^2-delta+13/4, 1/12*delta-1/2, 1/12*delta^2-delta+11/4], [-5/12*delta+5/2, 7/12*delta-7/2, -1/12*delta^2+7/6*delta-15/4, 1/12*delta^2-7/6*delta+47/12, 1/12*delta^2-7/6*delta+41/12, 1/12*delta-1/2, -1/12*delta^2+7/6*delta-15/4, -5/12*delta+5/2, 1/12*delta^2-7/6*delta+41/12, 1/12*delta-1/2, -1/12*delta^2+7/6*delta-13/4, 1/12*delta-1/2], [7/12*delta^2-7*delta+83/4, -5/12*delta^2+5*delta-59/4, 7/12*delta-7/2, -5/12*delta+5/2, -5/12*delta+5/2, -5/12*delta^2+5*delta-59/4, 1/12*delta-1/2, 1/12*delta^2-delta+11/4, 1/12*delta-1/2, 1/12*delta^2-delta+11/4, 1/12*delta-1/2, 1/12*delta^2-delta+13/4] ]):