  
  [1X4 [33X[0;0YIdempotents[133X[101X
  
  
  [1X4.1 [33X[0;0YComputing idempotents from character table[133X[101X
  
  [1X4.1-1 PrimitiveCentralIdempotentsByCharacterTable[101X
  
  [29X[2XPrimitiveCentralIdempotentsByCharacterTable[102X( [3XFG[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YA list of group algebra elements.[133X
  
  [33X[0;0YThe input [3XFG[103X should be a semisimple group algebra.[133X
  
  [33X[0;0YReturns  the list of primitive central idempotents of [3XFG[103X using the character
  table of [22XG[122X ([14X9.4[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XQS3 := GroupRing( Rationals, SymmetricGroup(3) );;                 [127X[104X
    [4X[25Xgap>[125X [27XPrimitiveCentralIdempotentsByCharacterTable( QS3 );[127X[104X
    [4X[28X[ (1/6)*()+(-1/6)*(2,3)+(-1/6)*(1,2)+(1/6)*(1,2,3)+(1/6)*(1,3,2)+(-1/6)*(1,3),[128X[104X
    [4X[28X  (2/3)*()+(-1/3)*(1,2,3)+(-1/3)*(1,3,2), (1/6)*()+(1/6)*(2,3)+(1/6)*(1,2)+(1/[128X[104X
    [4X[28X    6)*(1,2,3)+(1/6)*(1,3,2)+(1/6)*(1,3) ][128X[104X
    [4X[25Xgap>[125X [27XQG:=GroupRing( Rationals , SmallGroup(24,3) );[127X[104X
    [4X[28X<algebra-with-one over Rationals, with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XFG:=GroupRing( CF(3) , SmallGroup(24,3) );[127X[104X
    [4X[28X<algebra-with-one over CF(3), with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XpciQG := PrimitiveCentralIdempotentsByCharacterTable(QG);;[127X[104X
    [4X[25Xgap>[125X [27XpciFG := PrimitiveCentralIdempotentsByCharacterTable(FG);;[127X[104X
    [4X[25Xgap>[125X [27XLength(pciQG);[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27XLength(pciFG);[127X[104X
    [4X[28X7[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X4.2 [33X[0;0YTesting lists of idempotents for completeness[133X[101X
  
  [1X4.2-1 IsCompleteSetOfOrthogonalIdempotents[101X
  
  [29X[2XIsCompleteSetOfOrthogonalIdempotents[102X( [3XR[103X, [3Xlist[103X ) [32X operation
  
  [33X[0;0YThe input should be formed by a unital ring [3XR[103X and a list [3Xlist[103X of elements of
  [3XR[103X.[133X
  
  [33X[0;0YReturns  [9Xtrue[109X  if the list [3Xlist[103X is a complete list of orthogonal idempotents
  of  [3XR[103X.  That  is,  the  output is [9Xtrue[109X provided the following conditions are
  satisfied:[133X
  
  [33X[0;0Y[22X⋅[122X The sum of the elements of [3Xlist[103X is the identity of [3XR[103X,[133X
  
  [33X[0;0Y[22X⋅[122X [22Xe^2=e[122X, for every [22Xe[122X in [3Xlist[103X and[133X
  
  [33X[0;0Y[22X⋅[122X [22Xe*f=0[122X, if [22Xe[122X and [22Xf[122X are elements in different positions of [3Xlist[103X.[133X
  
  [33X[0;0YNo claim is made on the idempotents being central or primitive.[133X
  
  [33X[0;0YNote  that  the  if  a  non-zero  element  [22Xt[122X  of  [3XR[103X appears in two different
  positions  of [3Xlist[103X then the output is [9Xfalse[109X, and that the list [3Xlist[103X must not
  contain zeroes.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XQS5 := GroupRing( Rationals, SymmetricGroup(5) );;[127X[104X
    [4X[25Xgap>[125X [27Xidemp := PrimitiveCentralIdempotentsByCharacterTable( QS5 );;[127X[104X
    [4X[25Xgap>[125X [27XIsCompleteSetOfOrthogonalIdempotents( QS5, idemp );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCompleteSetOfOrthogonalIdempotents( QS5, [ One( QS5 ) ] );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCompleteSetOfOrthogonalIdempotents( QS5, [ One( QS5 ), One( QS5 ) ] );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X4.3 [33X[0;0YIdempotents from Shoda pairs[133X[101X
  
  [1X4.3-1 PrimitiveCentralIdempotentsByStrongSP[101X
  
  [29X[2XPrimitiveCentralIdempotentsByStrongSP[102X( [3XFG[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YA list of group algebra elements.[133X
  
  [33X[0;0YThe  input [3XFG[103X should be a semisimple group algebra of a finite group [22XG[122X whose
  coefficient field [22XF[122X is either a finite field or the field [22Xℚ[122X of rationals.[133X
  
  [33X[0;0YIf [22XF = ℚ[122X then the output is the list of primitive central idempotents of the
  group algebra [3XFG[103X realizable by strong Shoda pairs ([14X9.15[114X) of [22XG[122X.[133X
  
  [33X[0;0YIf  [22XF[122X  is  a  finite  field then the output is the list of primitive central
  idempotents  of  [3XFG[103X  realizable  by  strong  Shoda  pairs  [22X(K,H)[122X  of  [22XG[122X  and
  [22Xq[122X-cyclotomic classes modulo the index of [22XH[122X in [22XK[122X ([14X9.17[114X).[133X
  
  [33X[0;0YIf  the  list  of  primitive  central idempotents given by the output is not
  complete  (i.e.  if  the  group  [22XG[122X  is  not [13Xstrongly monomial[113X ([14X9.16[114X)) then a
  warning is displayed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XQG:=GroupRing( Rationals, AlternatingGroup(4) );;           [127X[104X
    [4X[25Xgap>[125X [27XPrimitiveCentralIdempotentsByStrongSP( QG );[127X[104X
    [4X[28X[ (1/12)*()+(1/12)*(2,3,4)+(1/12)*(2,4,3)+(1/12)*(1,2)(3,4)+(1/12)*(1,2,3)+(1/[128X[104X
    [4X[28X    12)*(1,2,4)+(1/12)*(1,3,2)+(1/12)*(1,3,4)+(1/12)*(1,3)(2,4)+(1/12)*[128X[104X
    [4X[28X    (1,4,2)+(1/12)*(1,4,3)+(1/12)*(1,4)(2,3),[128X[104X
    [4X[28X  (1/6)*()+(-1/12)*(2,3,4)+(-1/12)*(2,4,3)+(1/6)*(1,2)(3,4)+(-1/12)*(1,2,3)+([128X[104X
    [4X[28X    -1/12)*(1,2,4)+(-1/12)*(1,3,2)+(-1/12)*(1,3,4)+(1/6)*(1,3)(2,4)+(-1/12)*[128X[104X
    [4X[28X    (1,4,2)+(-1/12)*(1,4,3)+(1/6)*(1,4)(2,3),[128X[104X
    [4X[28X  (3/4)*()+(-1/4)*(1,2)(3,4)+(-1/4)*(1,3)(2,4)+(-1/4)*(1,4)(2,3) ][128X[104X
    [4X[25Xgap>[125X [27XQG := GroupRing( Rationals, SmallGroup(24,3) );;[127X[104X
    [4X[25Xgap>[125X [27XPrimitiveCentralIdempotentsByStrongSP( QG );;[127X[104X
    [4X[28XWedderga: Warning!!![128X[104X
    [4X[28XThe output is a NON-COMPLETE list of prim. central idemp.s of the input! [128X[104X
    [4X[25Xgap>[125X [27XFG := GroupRing( GF(2), Group((1,2,3)) );;[127X[104X
    [4X[25Xgap>[125X [27XPrimitiveCentralIdempotentsByStrongSP( FG );[127X[104X
    [4X[28X[ (Z(2)^0)*()+(Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,3,2), [128X[104X
    [4X[28X  (Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,3,2) ][128X[104X
    [4X[25Xgap>[125X [27XFG := GroupRing( GF(5), SmallGroup(24,3) );; [127X[104X
    [4X[25Xgap>[125X [27XPrimitiveCentralIdempotentsByStrongSP( FG );;[127X[104X
    [4X[28XWedderga: Warning!!![128X[104X
    [4X[28XThe output is a NON-COMPLETE list of prim. central idemp.s of the input! [128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.3-2 PrimitiveCentralIdempotentsBySP[101X
  
  [29X[2XPrimitiveCentralIdempotentsBySP[102X( [3XQG[103X ) [32X function
  [6XReturns:[106X  [33X[0;10YA list of group algebra elements.[133X
  
  [33X[0;0YThe input should be a rational group algebra of a finite group [22XG[122X.[133X
  
  [33X[0;0YReturns  a  list  containing  all the primitive central idempotents [22Xe[122X of the
  rational  group  algebra [3XQG[103X such that [22Xχ(e)ne 0[122X for some irreducible monomial
  character [22Xχ[122X of [22XG[122X.[133X
  
  [33X[0;0YThe  output  is  the  list of all primitive central idempotents of [3XQG[103X if and
  only if [22XG[122X is monomial, otherwise a warning message is displayed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XQG := GroupRing( Rationals, SymmetricGroup(4) );[127X[104X
    [4X[28X<algebra-with-one over Rationals, with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xpci:=PrimitiveCentralIdempotentsBySP( QG );[127X[104X
    [4X[28X[ (1/24)*()+(1/24)*(3,4)+(1/24)*(2,3)+(1/24)*(2,3,4)+(1/24)*(2,4,3)+(1/24)*[128X[104X
    [4X[28X    (2,4)+(1/24)*(1,2)+(1/24)*(1,2)(3,4)+(1/24)*(1,2,3)+(1/24)*(1,2,3,4)+(1/[128X[104X
    [4X[28X    24)*(1,2,4,3)+(1/24)*(1,2,4)+(1/24)*(1,3,2)+(1/24)*(1,3,4,2)+(1/24)*[128X[104X
    [4X[28X    (1,3)+(1/24)*(1,3,4)+(1/24)*(1,3)(2,4)+(1/24)*(1,3,2,4)+(1/24)*(1,4,3,2)+([128X[104X
    [4X[28X    1/24)*(1,4,2)+(1/24)*(1,4,3)+(1/24)*(1,4)+(1/24)*(1,4,2,3)+(1/24)*(1,4)[128X[104X
    [4X[28X    (2,3), (1/24)*()+(-1/24)*(3,4)+(-1/24)*(2,3)+(1/24)*(2,3,4)+(1/24)*[128X[104X
    [4X[28X    (2,4,3)+(-1/24)*(2,4)+(-1/24)*(1,2)+(1/24)*(1,2)(3,4)+(1/24)*(1,2,3)+(-1/[128X[104X
    [4X[28X    24)*(1,2,3,4)+(-1/24)*(1,2,4,3)+(1/24)*(1,2,4)+(1/24)*(1,3,2)+(-1/24)*[128X[104X
    [4X[28X    (1,3,4,2)+(-1/24)*(1,3)+(1/24)*(1,3,4)+(1/24)*(1,3)(2,4)+(-1/24)*[128X[104X
    [4X[28X    (1,3,2,4)+(-1/24)*(1,4,3,2)+(1/24)*(1,4,2)+(1/24)*(1,4,3)+(-1/24)*(1,4)+([128X[104X
    [4X[28X    -1/24)*(1,4,2,3)+(1/24)*(1,4)(2,3), (3/8)*()+(-1/8)*(3,4)+(-1/8)*(2,3)+([128X[104X
    [4X[28X    -1/8)*(2,4)+(-1/8)*(1,2)+(-1/8)*(1,2)(3,4)+(1/8)*(1,2,3,4)+(1/8)*[128X[104X
    [4X[28X    (1,2,4,3)+(1/8)*(1,3,4,2)+(-1/8)*(1,3)+(-1/8)*(1,3)(2,4)+(1/8)*(1,3,2,4)+([128X[104X
    [4X[28X    1/8)*(1,4,3,2)+(-1/8)*(1,4)+(1/8)*(1,4,2,3)+(-1/8)*(1,4)(2,3), [128X[104X
    [4X[28X  (3/8)*()+(1/8)*(3,4)+(1/8)*(2,3)+(1/8)*(2,4)+(1/8)*(1,2)+(-1/8)*(1,2)(3,4)+([128X[104X
    [4X[28X    -1/8)*(1,2,3,4)+(-1/8)*(1,2,4,3)+(-1/8)*(1,3,4,2)+(1/8)*(1,3)+(-1/8)*(1,3)[128X[104X
    [4X[28X    (2,4)+(-1/8)*(1,3,2,4)+(-1/8)*(1,4,3,2)+(1/8)*(1,4)+(-1/8)*(1,4,2,3)+(-1/[128X[104X
    [4X[28X    8)*(1,4)(2,3), (1/6)*()+(-1/12)*(2,3,4)+(-1/12)*(2,4,3)+(1/6)*(1,2)(3,4)+([128X[104X
    [4X[28X    -1/12)*(1,2,3)+(-1/12)*(1,2,4)+(-1/12)*(1,3,2)+(-1/12)*(1,3,4)+(1/6)*(1,3)[128X[104X
    [4X[28X    (2,4)+(-1/12)*(1,4,2)+(-1/12)*(1,4,3)+(1/6)*(1,4)(2,3) ][128X[104X
    [4X[25Xgap>[125X [27XIsCompleteSetOfPCIs(QG,pci);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XQS5 := GroupRing( Rationals, SymmetricGroup(5) );;[127X[104X
    [4X[25Xgap>[125X [27Xpci:=PrimitiveCentralIdempotentsBySP( QS5 );;[127X[104X
    [4X[28XWedderga: Warning!![128X[104X
    [4X[28XThe output is a NON-COMPLETE list of prim. central idemp.s of the input![128X[104X
    [4X[25Xgap>[125X [27XIsCompleteSetOfPCIs( QS5 , pci );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe   output  of  [2XPrimitiveCentralIdempotentsBySP[102X  contains  the  output  of
  [2XPrimitiveCentralIdempotentsByStrongSP[102X ([14X4.3-1[114X), possibly properly.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XQG := GroupRing( Rationals, SmallGroup(48,28) );;[127X[104X
    [4X[25Xgap>[125X [27Xpci:=PrimitiveCentralIdempotentsBySP( QG );;[127X[104X
    [4X[28XWedderga: Warning!![128X[104X
    [4X[28XThe output is a NON-COMPLETE list of prim. central idemp.s of the input! [128X[104X
    [4X[25Xgap>[125X [27XLength(pci);    [127X[104X
    [4X[28X6[128X[104X
    [4X[25Xgap>[125X [27Xspci:=PrimitiveCentralIdempotentsByStrongSP( QG );;  [127X[104X
    [4X[28XWedderga: Warning!!![128X[104X
    [4X[28XThe output is a NON-COMPLETE list of prim. central idemp.s of the input! [128X[104X
    [4X[25Xgap>[125X [27XLength(spci);[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27XIsSubset(pci,spci);          [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XQG:=GroupRing(Rationals,SmallGroup(1000,86));[127X[104X
    [4X[28X<algebra-with-one over Rationals, with 6 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsCompleteSetOfPCIs( QG , PrimitiveCentralIdempotentsBySP(QG) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCompleteSetOfPCIs( QG , PrimitiveCentralIdempotentsByStrongSP(QG) );[127X[104X
    [4X[28XWedderga: Warning!!![128X[104X
    [4X[28XThe output is a NON-COMPLETE list of prim. central idemp.s of the input![128X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X4.4  [33X[0;0YComplete  set  of orthogonal primitive idempotents from Shoda pairs and[101X
  [1Xcyclotomic classes[133X[101X
  
  [1X4.4-1 PrimitiveIdempotentsNilpotent[101X
  
  [29X[2XPrimitiveIdempotentsNilpotent[102X( [3XFG[103X, [3XH[103X, [3XK[103X, [3XC[103X, [3Xargs[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YA list of orthogonal primitive idempotents.[133X
  
  [33X[0;0YThe  input  [3XFG[103X  should  be  a semisimple group algebra of a finite nilpotent
  group  [22XG[122X  whose coefficient field [22XF[122X is a finite field. [3XH[103X and [3XK[103X should form a
  strong  Shoda  pair [22X(H,K)[122X of [22XG[122X. [3Xargs[103X is a list containing an epimorphism map
  [3Xepi[103X  from  [22XN_G(K)[122X  to  [22XN_G(K)/K[122X  and  a  generator  [3Xgq[103X  of  [22XH/K[122X.  [22XC[122X  is  the
  [22X|F|[122X-cyclotomic class modulo [22X[H:K][122X (w.r.t. the generator [22Xgq[122X of [22XH/K[122X)[133X
  
  [33X[0;0YThe  output  is  a  complete  set of orthogonal primitive idempotents of the
  simple algebra [22XFGe_C(G,H,K)[122X ([14X9.20[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XG:=DihedralGroup(8);; [127X[104X
    [4X[25Xgap>[125X [27XF:=GF(3);;                     [127X[104X
    [4X[25Xgap>[125X [27XFG:=GroupRing(F,G);;[127X[104X
    [4X[25Xgap>[125X [27XH:=StrongShodaPairs(G)[5][1];[127X[104X
    [4X[28XGroup([ f1*f2, f3, f3 ])[128X[104X
    [4X[25Xgap>[125X [27XK:=StrongShodaPairs(G)[5][2];[127X[104X
    [4X[28XGroup([ f1*f2 ])[128X[104X
    [4X[25Xgap>[125X [27XN:=Normalizer(G,K); [127X[104X
    [4X[28XGroup([ f1*f2*f3, f3 ])[128X[104X
    [4X[25Xgap>[125X [27Xepi:=NaturalHomomorphismByNormalSubgroup(N,K);[127X[104X
    [4X[28X[ f1*f2*f3, f3 ] -> [ f1, f1 ][128X[104X
    [4X[25Xgap>[125X [27XQHK:=Image(epi,H); [127X[104X
    [4X[28XGroup([ <identity> of ..., f1, f1 ])[128X[104X
    [4X[25Xgap>[125X [27Xgq:=MinimalGeneratingSet(QHK)[1]; [127X[104X
    [4X[28Xf1[128X[104X
    [4X[25Xgap>[125X [27XC:=CyclotomicClasses(Size(F),Index(H,K))[2];[127X[104X
    [4X[28X[ 1 ][128X[104X
    [4X[25Xgap>[125X [27XPrimitiveIdempotentsNilpotent(FG,H,K,C,[epi,gq]);[127X[104X
    [4X[28X[ (Z(3)^0)*<identity> of ...+(Z(3))*f3+(Z(3)^0)*f1*f2+(Z(3))*f1*f2*f3, [128X[104X
    [4X[28X  (Z(3)^0)*<identity> of ...+(Z(3))*f3+(Z(3))*f1*f2+(Z(3)^0)*f1*f2*f3 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.4-2 PrimitiveIdempotentsTrivialTwisting[101X
  
  [29X[2XPrimitiveIdempotentsTrivialTwisting[102X( [3XFG[103X, [3XH[103X, [3XK[103X, [3XC[103X, [3Xargs[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YA list of orthogonal primitive idempotents.[133X
  
  [33X[0;0YThe  input [3XFG[103X should be a semisimple group algebra of a finite group [22XG[122X whose
  coefficient  field  [22XF[122X  is a finite field. [3XH[103X and [3XK[103X should form a strong Shoda
  pair  [22X(H,K)[122X  of  [22XG[122X.  [3Xargs[103X  is  a list containing an epimorphism map [3Xepi[103X from
  [22XN_G(K)[122X  to [22XN_G(K)/K[122X and a generator [3Xgq[103X of [22XH/K[122X. [22XC[122X is the [22X|F|[122X-cyclotomic class
  modulo  [22X[H:K][122X  (w.r.t. the generator [22Xgq[122X of [22XH/K[122X). The input parameters should
  be such that the simple component [22XFGe_C(G,H,K)[122X has a trivial twisting.[133X
  
  [33X[0;0YThe  output  is  a  complete  set of orthogonal primitive idempotents of the
  simple algebra [22XFGe_C(G,H,K)[122X ([14X9.20[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XG:=DihedralGroup(8);; [127X[104X
    [4X[25Xgap>[125X [27XF:=GF(3);;                     [127X[104X
    [4X[25Xgap>[125X [27XFG:=GroupRing(F,G);;[127X[104X
    [4X[25Xgap>[125X [27XH:=StrongShodaPairs(G)[5][1];[127X[104X
    [4X[28XGroup([ f1*f2, f3, f3 ])[128X[104X
    [4X[25Xgap>[125X [27XK:=StrongShodaPairs(G)[5][2];[127X[104X
    [4X[28XGroup([ f1*f2 ])[128X[104X
    [4X[25Xgap>[125X [27XN:=Normalizer(G,K); [127X[104X
    [4X[28XGroup([ f1*f2*f3, f3 ])[128X[104X
    [4X[25Xgap>[125X [27Xepi:=NaturalHomomorphismByNormalSubgroup(N,K);[127X[104X
    [4X[28X[ f1*f2*f3, f3 ] -> [ f1, f1 ][128X[104X
    [4X[25Xgap>[125X [27XQHK:=Image(epi,H); [127X[104X
    [4X[28XGroup([ <identity> of ..., f1, f1 ])[128X[104X
    [4X[25Xgap>[125X [27Xgq:=MinimalGeneratingSet(QHK)[1]; [127X[104X
    [4X[28Xf1[128X[104X
    [4X[25Xgap>[125X [27XC:=CyclotomicClasses(Size(F),Index(H,K))[2];[127X[104X
    [4X[28X[ 1 ][128X[104X
    [4X[25Xgap>[125X [27XPrimitiveIdempotentsTrivialTwisting(FG,H,K,C,[epi,gq]);[127X[104X
    [4X[28X[ (Z(3)^0)*<identity> of ...+(Z(3))*f3+(Z(3)^0)*f1*f2+(Z(3))*f1*f2*f3, [128X[104X
    [4X[28X  (Z(3)^0)*<identity> of ...+(Z(3))*f3+(Z(3))*f1*f2+(Z(3)^0)*f1*f2*f3 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
