% This is a file in REDUCE syntax. Read it in to REDUCE by
% typing 
% in filename$
% or 
% in filename;
% (the former just gives the output; the former gives the input too).
% REDUCE reads until it encounters the line
% ;end;

on nat;       % Replace this by off nat;  if you want printable output.

% Let $z$ be a primitive 7-th root of 1:

let z^6=-(z^5+z^4+z^3+z^2+z+1);
% The following is a square root of -7:

s:=1+2*z+2*z^2+2*z^4$

% The following are zero:
write "The following is zero, checking that s is a square root of -7:";

s^2+7;   

write "The following is zero, checking that z satisfies a cubic polynomial over \Q(s):";
z^3-((s-1)*z^2+(s+1)*z+2)/2;

phiz:=z^2$
phi2z:=z^4$

ID3:=mat((1,0,0),
         (0,1,0),
         (0,0,1))$
% The following element generates, together with $-1$,
% the set of $x$ in $\ell=\Q(s)$ such that $v(x)=0$
% for all valuations $v$ on~$\ell$ except the 2-adic
% ones, and such that ${\bar x}x=1$.

gendt:=(3+s)/4$
gendti:=(3-s)/4$

SIG:=mat((      0,1,0),
         (      0,0,1),
         ((3+s)/4,0,0))$

% Inverse of SIG:
SIGI:=((3-s)/4)*SIG^2$

% The following is zero:
% SIG*SIGI-ID3;

ZMAT:=mat((z,  0,  0),
          (0,z^2,  0),
          (0,  0,z^4))$

ZMATI:=mat((z^6,  0,  0),
           (  0,z^5,  0),
           (  0,  0,z^3))$

% w:=z+z^6$
% phiw:=phiz+phiz^6$
% phi2w:=phi2z+phi2z^6$
% 
% FF:=mat(
% (w,   0,    0),
% (0,phiw,    0),
% (0,   0,phi2w))$

% Explicitly:
FF:=mat(
(-(z^5+z^4+z^3+z^2+1),0,0),
(0,z^2*(z^3+1),0),
(0,0,z^3*(z+1)))$

% Inverse of w:
% wi:=w^2+w-2$
% phiwi:=phiw^2+phiw-2$
% phi2wi:=phi2w^2+phi2w-2$
% 
% Here is the inverse of FF:
% 
% FFI:=mat((wi,   0,    0),
%           (0,phiwi,    0),
%           (0,   0,phi2wi))$
% 
% Explicitly,
FFI:=mat(
(-(z^4+z^3+1),0,0),
(0,z^2*(z^3+z^2+z+1),0),
(0,0,-(z^5+z^2+1)))$

ID18:=mat(
(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1))$

% Here are the various condition matrices:

CondMtxDM7Type1:=mat(
(9,1,3,9,1,28,34,1,3,43,0,6,9,46,3,24,47,30),
(43,0,6,43,0,40,21,0,6,31,2,28,43,1,6,6,48,43),
(0,0,0,18,1,40,22,2,3,15,47,21,9,46,6,18,48,21),
(0,0,0,37,0,31,18,0,6,28,3,25,43,0,12,21,47,9),
(0,0,0,15,0,27,34,48,0,15,46,24,0,1,22,10,4,22),
(0,0,0,12,0,37,37,2,0,12,48,15,0,47,12,15,48,12),
(0,0,25,25,0,24,9,48,3,15,46,46,15,1,27,44,4,44),
(0,0,34,27,0,15,10,2,22,12,48,27,12,47,37,3,48,24),
(18,1,40,4,1,12,42,45,18,43,48,34,9,2,15,47,4,35),
(37,0,31,30,0,24,10,3,19,15,46,36,27,47,46,28,1,6),
(15,0,27,19,48,22,30,47,24,34,4,47,10,3,0,14,44,30),
(12,0,37,25,2,12,24,46,15,37,48,46,15,1,0,7,3,10),
(12,0,24,16,1,22,30,47,30,0,0,31,34,1,9,3,1,40),
(3,0,31,0,0,12,40,1,43,0,48,46,21,48,18,25,1,31),
(15,0,27,22,1,19,24,46,33,9,1,31,13,2,43,24,48,3),
(12,0,37,34,1,6,6,0,0,43,48,46,24,48,21,22,2,22),
(22,1,6,36,46,9,22,2,28,12,1,9,31,0,31,27,47,33),
(18,0,12,25,1,34,18,47,24,3,0,34,28,2,46,15,48,0))$

CondMtxDM7Type2:=mat(
(9,1,3,9,1,28,34,1,3,43,0,6,9,46,3,24,47,30),
(43,0,6,43,0,40,21,0,6,31,2,28,43,1,6,6,48,43),
(0,0,0,30,48,32,4,47,22,20,39,26,15,3,44,45,9,41),
(0,0,0,24,2,13,13,4,12,9,48,18,12,43,24,8,45,2),
(0,0,0,15,0,27,34,48,0,15,46,24,0,1,22,10,4,22),
(0,0,0,12,0,37,37,2,0,12,48,15,0,47,12,15,48,12),
(0,0,37,12,0,12,28,1,34,9,48,15,9,48,46,43,0,6),
(0,0,9,3,0,40,3,0,3,43,1,46,43,0,43,15,48,12),
(18,1,40,4,1,12,42,45,18,43,48,34,9,2,15,47,4,35),
(37,0,31,30,0,24,10,3,19,15,46,36,27,47,46,28,1,6),
(9,0,46,31,1,3,18,47,12,40,0,40,3,1,0,34,1,31),
(43,0,43,12,0,6,37,1,40,6,48,15,9,48,0,37,1,46),
(12,0,24,16,1,22,30,47,30,0,0,31,34,1,9,3,1,40),
(3,0,31,0,0,12,40,1,43,0,48,46,21,48,18,25,1,31),
(35,0,21,38,44,6,1,3,16,15,3,30,38,2,20,35,42,7),
(42,0,42,29,3,44,5,43,17,12,1,10,1,3,9,21,0,28),
(22,1,6,36,46,9,22,2,28,12,1,9,31,0,31,27,47,33),
(18,0,12,25,1,34,18,47,24,3,0,34,28,2,46,15,48,0))$

CondMtxDM3Type1:=ID18$

CondMtxDM3Type2:=(1/3)*mat(
(18,3,15,12,0,0,9,0,15,24,3,21,12,24,21,15,24,0),
(15,0,15,24,0,9,9,0,21,6,0,12,12,3,12,24,0,21),
(21,0,24,18,3,18,24,0,0,6,0,3,21,24,18,18,24,6),
(12,0,24,21,0,0,15,0,24,21,3,15,12,0,0,9,0,6),
(4,0,23,5,0,25,6,1,16,15,26,18,13,26,21,3,0,23),
(13,0,25,26,0,13,6,0,22,23,1,25,26,0,15,3,26,5),
(12,0,0,9,0,15,24,3,21,12,24,21,15,24,0,18,0,9),
(24,0,9,9,0,21,6,0,12,12,3,12,24,0,21,18,24,18),
(18,3,18,24,0,0,6,0,3,21,24,18,18,24,6,0,3,12),
(21,0,0,15,0,24,21,3,15,12,0,0,9,0,6,18,24,12),
(5,0,25,6,1,16,15,26,18,13,26,21,3,0,23,8,1,9),
(26,0,13,6,0,22,23,1,25,26,0,15,3,26,5,11,0,3),
(0,0,18,18,9,9,9,18,9,18,18,0,0,0,0,0,9,18),
(0,0,9,18,0,9,9,9,9,18,0,9,0,18,0,18,0,18),
(18,0,0,18,0,9,9,18,0,0,18,18,0,9,9,9,0,18),
(18,0,18,9,9,18,9,0,0,0,0,18,0,18,9,9,0,18),
(18,3,21,18,24,0,12,24,9,9,0,15,24,3,0,12,0,15),
(18,0,12,15,3,21,24,0,18,9,24,15,6,0,9,12,0,21))$

CondMtxDM5Type1:=ID18$

CondMtxDM5Type2:=(1/5)*mat(
(40,5,50,75,0,105,35,0,50,30,5,5,40,120,40,50,120,35),
(30,0,65,30,0,90,20,0,95,45,0,60,115,5,115,10,0,75),
(95,0,30,105,5,35,10,0,110,15,0,110,60,120,70,55,120,105),
(115,0,65,95,0,70,65,0,85,75,5,30,60,0,0,90,0,45),
(8,0,10,60,0,1,96,1,108,8,124,7,115,124,68,57,0,85),
(48,0,94,77,0,87,0,0,48,31,1,40,56,0,0,29,124,38),
(75,0,105,35,0,50,30,5,5,40,120,40,50,120,35,105,0,90),
(30,0,90,20,0,95,45,0,60,115,5,115,10,0,75,0,120,0),
(105,5,35,10,0,110,15,0,110,60,120,70,55,120,105,35,5,40),
(95,0,70,65,0,85,75,5,30,60,0,0,90,0,45,0,120,80),
(60,0,1,96,1,108,8,124,7,115,124,68,57,0,85,31,1,96),
(77,0,87,0,0,48,31,1,40,56,0,0,29,124,38,9,0,68),
(50,0,0,25,25,25,75,100,75,0,100,50,25,0,75,75,25,0),
(100,0,100,100,0,50,75,25,75,50,0,0,0,100,0,25,0,75),
(50,0,50,75,0,50,50,100,100,25,100,25,50,25,75,100,0,25),
(75,0,50,0,25,25,50,0,0,75,0,100,0,100,25,75,0,75),
(105,5,40,40,120,35,75,120,90,35,0,50,30,5,105,40,0,50),
(0,0,115,30,5,75,30,0,0,20,120,65,45,0,90,115,0,95))$

;end;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generators for $\bar\Gamma$ in the case $(a=7,p=2,\emptyset)$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Firstly, the sequences of coefficients:

z1:=0$
z2:=0$
z3:=0$
z4:=0$
z5:=1$
z6:=0$
z7:=0$
z8:=0$
z9:=0$
z10:=0$
z11:=0$
z12:=0$
z13:=0$
z14:=0$
z15:=0$
z16:=0$
z17:=0$
z18:=0$

b1:=7/7$
b2:=0/7$
b3:=7/7$
b4:=6/7$
b5:=4/7$
b6:=5/7$
b7:=4/7$
b8:=5/7$
b9:=-6/7$
b10:=8/7$
b11:=3/7$
b12:=2/7$
b13:=11/7$
b14:=-2/7$
b15:=8/7$
b16:=-1/7$
b17:=-3/7$
b18:=-2/7$

% Here are the matrix forms of the elements of the division algebra 
% $\cD$ corresponding to the above sequences of coefficients:

write "Here is the matrix for the element BB of the division algebra:";
BB:=
 b1*ZMAT^0*SIGI +b2*ZMAT^0*ID3 +b3*ZMAT^0*SIG+
 b4*ZMAT^1*SIGI +b5*ZMAT^1*ID3 +b6*ZMAT^1*SIG+
 b7*ZMAT^2*SIGI +b8*ZMAT^2*ID3 +b9*ZMAT^2*SIG+
b10*ZMAT^3*SIGI+b11*ZMAT^3*ID3+b12*ZMAT^3*SIG+
b13*ZMAT^4*SIGI+b14*ZMAT^4*ID3+b15*ZMAT^4*SIG+
b16*ZMAT^5*SIGI+b17*ZMAT^5*ID3+b18*ZMAT^5*SIG;

write "Here is the matrix for the element ZZ of the division algebra:";
ZZ:=
 z1*ZMAT^0*SIGI +z2*ZMAT^0*ID3 +z3*ZMAT^0*SIG+
 z4*ZMAT^1*SIGI +z5*ZMAT^1*ID3 +z6*ZMAT^1*SIG+
 z7*ZMAT^2*SIGI +z8*ZMAT^2*ID3 +z9*ZMAT^2*SIG+
z10*ZMAT^3*SIGI+z11*ZMAT^3*ID3+z12*ZMAT^3*SIG+
z13*ZMAT^4*SIGI+z14*ZMAT^4*ID3+z15*ZMAT^4*SIG+
z16*ZMAT^5*SIGI+z17*ZMAT^5*ID3+z18*ZMAT^5*SIG;

% Of course, ZZ is simply ZMAT.
write "The following is zero, checking that ZZ equals ZMAT:";

ZZ-ZMAT;

ZZSTAR:=sub(z=z^6,tp(ZZ))$
BBSTAR:=sub(z=z^6,tp(BB))$

% Checking that the elements $\xi$ satisfy $\iota(\xi)\xi=1$:
% The following are zero:
write "The following are zero, checking that BB and ZZ are unitary with respect to FF:";

BBSTAR*FF*BB-FF;
ZZSTAR*FF*ZZ-FF;

% Therefore, the inverses of ZZ and BB can be calculated using
ZZI:=FFI*ZZSTAR*FF$
BBI:=FFI*BBSTAR*FF$

% Explicitly:
% BB:=(1/7)*mat(
% (-3*z^5-2*z^4+3*z^3+5*z^2+4*z,-2*z^5+8*z^4+2*z^3-6*z^2+5*z+7,-6*z^5+3*z^4+6*z^3+3*z^2+z+7),
% (-8*z^5-4*z^4-9*z^3-2*z^2+3*z-1,-3*z^5+2*z^4-6*z^3+z^2-5*z-3,-2*z^5-8*z^4-4*z^3+3*z^2+6*z+5),
% (1/2*(-8*z^5+14*z^4-11*z^3+8*z^2-13*z+3),9*z^5+7*z^4+z^3+12*z^2+5*z+8,6*z^5+7*z^4+3*z^3+z^2+8*z+3))$

% BBI:=(1/7)*mat(
% (z^5-z^4-6*z^3-7*z^2-4*z-4,-5*z^5-5*z^4+3*z^2-10*z-4,1/2*(7*z^5-12*z^4-8*z^3-2*z^2-z-5)),
% (3*z^5+4*z^4+3*z^3-5*z+2,6*z^5-z^4+7*z^3+2*z^2+5*z+2,3*z^4-5*z^3-10*z^2-5*z-4),
% (9*z^5-2*z^4+9*z^3+7*z^2+6*z+6,-3*z^5-3*z^4-8*z^2+z-1,-7*z^5-5*z^4-z^3-2*z^2-8*z-5))$

% ZZI:=mat(
% (z^6,  0,  0),
% (  0,z^5,  0),
% (  0,  0,z^3))$

% Relations:
% The following are zero:
write "The following are zero, checking various relations hold:";

ZZ^7 - ID3;
(BBI*BBI*ZZ^1)^3 - gendti^2*ID3;
(BB*BB*ZZ^5*BB*BB*ZZ^2)^3 - gendt^4*ID3;
(BB*BB*ZZ^5*BB*BB*ZZ^4)^3 - gendt^4*ID3;
BB*BB*BB*ZZ^5*BBI*ZZ^2*BBI*BBI*ZZ^1 - ID3;
BB*BB*BB*ZZ^1*BB*BB*BB*ZZ^3*BB*ZZ^2*BBI*ZZ^6 - gendt^2*ID3;
BB*BB*BB*ZZ^2*BB*BB*ZZ^5*BBI*ZZ^6*BBI*BBI*BBI*ZZ^1*BBI*ZZ^6 - ID3;

zvec:=mat((z1),(z2),(z3),(z4),(z5),(z6),(z7),(z8),(z9),(z10),(z11),(z12),(z13),(z14),(z15),(z16),(z17),(z18))$

bvec:=mat((b1),(b2),(b3),(b4),(b5),(b6),(b7),(b8),(b9),(b10),(b11),(b12),(b13),(b14),(b15),(b16),(b17),(b18))$

% The following are zero, showing that BB satisfies the type 1 3-adic, 5-adic and 7-adic conditions:

write "The following are zero, checking that BB and ZZ satisfy the integrality conditions:";

CondMtxDM3Type1*bvec - (1/7)*mat((7),(0),(7),(6),(4),(5),(4),(5),(-6),(8),(3),(2),(11),(-2),(8),(-1),(-3),(-2));
CondMtxDM5Type1*bvec - (1/7)*mat((7),(0),(7),(6),(4),(5),(4),(5),(-6),(8),(3),(2),(11),(-2),(8),(-1),(-3),(-2));
CondMtxDM7Type1*bvec - mat((81),(206),(78),(155),(145),(72),(184),(118),(203),(225),(171),(208),(151),(90),(157),(207),(149),(186));

% The following are zero, showing that ZZ satisfies the type 1 3-adic, 5-adic and 7-adic conditions:
CondMtxDM3Type1*zvec - mat((0),(0),(0),(0),(1),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0));
CondMtxDM5Type1*zvec - mat((0),(0),(0),(0),(1),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0));
CondMtxDM7Type1*zvec - mat((1),(0),(1),(0),(0),(0),(0),(0),(1),(0),(48),(2),(1),(0),(1),(1),(46),(1));

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generators for $\bar\Gamma$ in the case $(a=7,p=2,\{3\})$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Firstly, the sequences of coefficients:

a1:=-8/21$
a2:=-2/21$
a3:=14/21$
a4:=1/21$
a5:=-9/21$
a6:=27/21$
a7:=-8/21$
a8:=-14/21$
a9:=4/21$
a10:=-7/21$
a11:=-10/21$
a12:=15/21$
a13:=-3/21$
a14:=-4/21$
a15:=25/21$
a16:=-3/21$
a17:=4/21$
a18:=20/21$

b1:=1/3$
b2:=1/3$
b3:=-1/3$
b4:=1/3$
b5:=0/3$
b6:=-3/3$
b7:=-2/3$
b8:=1/3$
b9:=-2/3$
b10:=-1/3$
b11:=-1/3$
b12:=0/3$
b13:=0/3$
b14:=-1/3$
b15:=-2/3$
b16:=0/3$
b17:=-2/3$
b18:=-4/3$

c1:=-4/7$
c2:=0/7$
c3:=-5/7$
c4:=1/7$
c5:=6/7$
c6:=8/7$
c7:=8/7$
c8:=4/7$
c9:=4/7$
c10:=3/7$
c11:=8/7$
c12:=-3/7$
c13:=-7/7$
c14:=4/7$
c15:=1/7$
c16:=-1/7$
c17:=-1/7$
c18:=9/7$


% Here are the matrix forms of the elements of $\cD$ corresponding
% to the above sequences of coefficients:

write "Here is the matrix for the element AA of the division algebra:"$
AA:=
 a1*ZMAT^0*SIGI +a2*ZMAT^0*ID3 +a3*ZMAT^0*SIG+
 a4*ZMAT^1*SIGI +a5*ZMAT^1*ID3 +a6*ZMAT^1*SIG+
 a7*ZMAT^2*SIGI +a8*ZMAT^2*ID3 +a9*ZMAT^2*SIG+
a10*ZMAT^3*SIGI+a11*ZMAT^3*ID3+a12*ZMAT^3*SIG+
a13*ZMAT^4*SIGI+a14*ZMAT^4*ID3+a15*ZMAT^4*SIG+
a16*ZMAT^5*SIGI+a17*ZMAT^5*ID3+a18*ZMAT^5*SIG;

write "Here is the matrix for the element BB of the division algebra:"$
BB:=
 b1*ZMAT^0*SIGI +b2*ZMAT^0*ID3 +b3*ZMAT^0*SIG+
 b4*ZMAT^1*SIGI +b5*ZMAT^1*ID3 +b6*ZMAT^1*SIG+
 b7*ZMAT^2*SIGI +b8*ZMAT^2*ID3 +b9*ZMAT^2*SIG+
b10*ZMAT^3*SIGI+b11*ZMAT^3*ID3+b12*ZMAT^3*SIG+
b13*ZMAT^4*SIGI+b14*ZMAT^4*ID3+b15*ZMAT^4*SIG+
b16*ZMAT^5*SIGI+b17*ZMAT^5*ID3+b18*ZMAT^5*SIG;

write "Here is the matrix for the element CC of the division algebra:"$
CC:=
 c1*ZMAT^0*SIGI +c2*ZMAT^0*ID3 +c3*ZMAT^0*SIG+
 c4*ZMAT^1*SIGI +c5*ZMAT^1*ID3 +c6*ZMAT^1*SIG+
 c7*ZMAT^2*SIGI +c8*ZMAT^2*ID3 +c9*ZMAT^2*SIG+
c10*ZMAT^3*SIGI+c11*ZMAT^3*ID3+c12*ZMAT^3*SIG+
c13*ZMAT^4*SIGI+c14*ZMAT^4*ID3+c15*ZMAT^4*SIG+
c16*ZMAT^5*SIGI+c17*ZMAT^5*ID3+c18*ZMAT^5*SIG;


%% Here are AA, BB and CC, as matrices with entries in \Q(z):
%% AA:=(1/21)*mat(
%% (4*z^5-4*z^4-10*z^3-14*z^2-9*z-2,20*z^5+25*z^4+15*z^3+4*z^2+27*z+14,-4*z^5+3*z^4-7*z^3-6*z^2-z-6),
%% (7*z^5-z^4+4*z^3+8*z^2+4*z-1,10*z^5-4*z^4+14*z^3+z^2+6*z+8,-15*z^5-11*z^4+5*z^3+12*z^2+10*z-1),
%% (-23*z^5-7*z^4-29*z^3-12*z^2-19*z-22,-4*z^5+4*z^4+3*z^3-5*z-5,-14*z^5-13*z^4-4*z^3-8*z^2-18*z-6))$
%% 
%% BB:=(1/3)*mat(
%% (-2*z^5-z^4-z^3+z^2+1,-4*z^5-2*z^4-2*z^2-3*z-1,-z^5-z^3-3*z^2-z),
%% (z^5-z^4+z^3+2*z^2+z+2,z^5+2*z^4-z^3+z^2+2,-2*z^4-4*z^3-3*z^2-2*z-1),
%% ((8*z^5+4*z^4+5*z^3+6*z^2+z+7)/2,-z^5+z^4-2*z+1,z^5+2*z^4+2*z^3+z^2+3*z+3))$
%% 
%% CC:=(1/7)*mat(
%% (-z^5+4*z^4+8*z^3+4*z^2+6*z,9*z^5+z^4-3*z^3+4*z^2+8*z-5,z^5-7*z^4-3*z^3+6*z^2+6*z-3),
%% (-3*z^5+5*z^4-4*z^3-2*z^2-10*z-7,-8*z^5-4*z^4-9*z^3-2*z^2-4*z-8,3*z^5+7*z^4+12*z^3+11*z^2+4*z-2),
%% (-9*z^5-6*z^4-5*z^3-13*z^2-2*z-14,4*z^5+2*z^4+z^3-6*z^2+9*z-3,9*z^5+7*z^4+z^3+5*z^2+5*z+1))$
 
AI:=(1/21)*mat(
(-5*z^5-z^4+5*z^3+13*z^2+9*z+7,11*z^5-14*z^4-12*z^3+10*z^2+3*z-19,-7*z^5-3*z^4+5*z^3-18*z^2-16*z-3),
(-2*z^5-19*z^4-2*z^3-7*z^2+z+8,-5*z^5+8*z^4-10*z^3+4*z^2-6*z+2,12*z^5+22*z^4+23*z^3+15*z^2-2*z-7),
(-5*z^5-13*z^4+4*z^3-24*z^2+8*z-19,2*z^5-5*z^4+3*z^2-17*z+10,10*z^5+14*z^4+5*z^3+4*z^2+18*z+12))$

BI:=(1/3)*mat(
(z^5-z^4-z^3-2*z^2+1,2*z^5+z^4+z^2+3*z-1,1/2*(z^5-2*z^3+z+3)),
(z^5+2*z^4+z^3+2*z^2+z+2,z^5-z^4+2*z^3+z^2+2,z^4+2*z^3+3*z^2+z-1),
(-2*z^5-z^4-2*z^3-3*z^2-z-4,-z^5+z^4+z+1,-2*z^5-z^4-z^3-2*z^2-3*z))$

CI:=(1/7)*mat(
(-2*z^5+2*z^4-2*z^3-7*z^2-6*z-6,z^5+6*z^4-6*z^3+3*z+3,-4*z^5-3*z^4-4*z^3-7*z^2-5*z-5),
(3*z^5-4*z^4+z^2-z+1,2*z^5-5*z^4-4*z^2+4*z-4,6*z^5+6*z^4+7*z^3+9*z^2+12*z+9),
(-7*z^5+5*z^4+z^3+2*z^2+z-2,z^4+3*z^3-z^2-4*z+1,-4*z^4+2*z^3+4*z^2-5*z-4))$

AASTAR:=sub(z=z^6,tp(AA))$
BBSTAR:=sub(z=z^6,tp(BB))$
CCSTAR:=sub(z=z^6,tp(CC))$

% Checking that the elements $\xi$ satisfy $\iota(\xi)\xi=1$:
% The following are zero:
write "The following are zero, checking that AA, BB and CC are unitary with respect to FF:";

AASTAR*FF*AA-FF;
BBSTAR*FF*BB-FF;
CCSTAR*FF*CC-FF;

% Therefore, the inverses of AA, BB and CC can be calculated using
AAI:=FFI*AASTAR*FF$
BBI:=FFI*BBSTAR*FF$
CCI:=FFI*CCSTAR*FF$

avec:=mat((a1),(a2),(a3),(a4),(a5),(a6),(a7),(a8),(a9),(a10),(a11),(a12),(a13),(a14),(a15),(a16),(a17),(a18))$
bvec:=mat((b1),(b2),(b3),(b4),(b5),(b6),(b7),(b8),(b9),(b10),(b11),(b12),(b13),(b14),(b15),(b16),(b17),(b18))$
cvec:=mat((c1),(c2),(c3),(c4),(c5),(c6),(c7),(c8),(c9),(c10),(c11),(c12),(c13),(c14),(c15),(c16),(c17),(c18))$

% Checking the integality conditions: The following are zero:

CondMtxDM3Type2*avec-(1/7)*mat((69),(123),(101),(38),(190),(151),(16),(159),(25),(74),(113),(62),(26),(52),(45),(134),(-12),(108));
CondMtxDM3Type2*bvec-mat((-19),(-20),(-25),(-13),(-37),(-23),(-23),(-28),(-14),(-25),(-25),(-24),(-19),(-17),(-12),(-19),(-8),(-29));
CondMtxDM3Type2*cvec-(1/7)*mat((8),(68),(52),(24),(209),(-20),(196),(43),(68),(136),(183),(146),(195),(144),(150),(99),(54),(217));

CondMtxDM5Type1*avec-(1/21)*mat((-8),(-2),(14),(1),(-9),(27),(-8),(-14),(4),(-7),(-10),(15),(-3),(-4),(25),(-3),(4),(20));
CondMtxDM5Type1*bvec-(1/3)*mat((1),(1),(-1),(1),(0),(-3),(-2),(1),(-2),(-1),(-1),(0),(0),(-1),(-2),(0),(-2),(-4));
CondMtxDM5Type1*cvec-(1/7)*mat((-4),(0),(-5),(1),(6),(8),(8),(4),(4),(3),(8),(-3),(-7),(4),(1),(-1),(-1),(9));

CondMtxDM7Type1*avec-(1/3)*mat((117),(279),(162),(197),(81),(121),(258),(299),(69),(238),(108),(10),(206),(218),(240),(159),(167),(278));
CondMtxDM7Type1*bvec-(1/3)*mat((-450),(-408),(-450),(-289),(-288),(-446),(-348),(-487),(-399),(-317),(-369),(-200),(-316),(-487),(-321),(-294),(-373),(-301));
CondMtxDM7Type1*cvec-mat((130),(60),(164),(30),(178),(176),(131),(114),(151),(66),(137),(96),(88),(141),(50),(71),(88),(43));

% Relations: The following are zero:

(BB*AA^2*BB)^3 - gendt^2*ID3;
(BB*CC*BI*AA*BI*CC)^3-gendti*ID3;
(AA^2*BB*AI*BI*CC*BI*CC)^3 - gendti*ID3;
CI*BB*CI*AA^2*BB*AA*BB - gendt*ID3;
CC^2*BI*CI*BB*AI*BI*AI*BI*AI*CC*BI - gendti*ID3;
CI*AA^2*BB*AI*BI*AA*BI*CC*AA*BB*AA - ID3;
BB*CI*BB*AA*BB*AA*CI*BB*AI*CC*BI*CI - gendt*ID3;
BI*AA*BI*CC*AI*BI^2*AI^2*CI^2*BB*CI*AI - gendti*ID3;
AI*BB*AA*CI*BB*AI*BB^2*AI*CC*BI*AA*BI*CC*BB*CC - gendt*ID3;
CI*BB*AI*CC*BI*CC^2*AI*BI*AI^2*BI*CI*BB*AI*BB*CI*AA - ID3;
AA*BB*AA*BB*CC*BI*AA*CI*AA^2*BB*AI*BB*CI*BB*AI*CC*BI*CC^2 - gendt*ID3;


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generators for $\bar\Gamma$ in the case $(a=7,p=2,\{5\})$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Firstly, the sequences of coefficients:

a1:=-2/5$
a2:=0/5$
a3:=0/5$
a4:=-6/5$
a5:=-1/5$
a6:=-4/5$
a7:=-2/5$
a8:=4/5$
a9:=0/5$
a10:=2/5$
a11:=4/5$
a12:=2/5$
a13:=0/5$
a14:=2/5$
a15:=-4/5$
a16:=-2/5$
a17:=6/5$
a18:=-2/5$

b1:=2/7$
b2:=0/7$
b3:=-1/7$
b4:=6/7$
b5:=-1/7$
b6:=-5/7$
b7:=5/7$
b8:=-3/7$
b9:=-5/7$
b10:=-1/7$
b11:=-6/7$
b12:=-1/7$
b13:=2/7$
b14:=-3/7$
b15:=-7/7$
b16:=7/7$
b17:=6/7$
b18:=-2/7$

% Here are the matrix forms of the elements of $\cD$ corresponding
% to the above sequences of coefficients:

write "Here is the matrix for the element AA of the division algebra:"$
AA:=
 a1*ZMAT^0*SIGI +a2*ZMAT^0*ID3 +a3*ZMAT^0*SIG+
 a4*ZMAT^1*SIGI +a5*ZMAT^1*ID3 +a6*ZMAT^1*SIG+
 a7*ZMAT^2*SIGI +a8*ZMAT^2*ID3 +a9*ZMAT^2*SIG+
a10*ZMAT^3*SIGI+a11*ZMAT^3*ID3+a12*ZMAT^3*SIG+
a13*ZMAT^4*SIGI+a14*ZMAT^4*ID3+a15*ZMAT^4*SIG+
a16*ZMAT^5*SIGI+a17*ZMAT^5*ID3+a18*ZMAT^5*SIG;

write "Here is the matrix for the element BB of the division algebra:"$
BB:=
 b1*ZMAT^0*SIGI +b2*ZMAT^0*ID3 +b3*ZMAT^0*SIG+
 b4*ZMAT^1*SIGI +b5*ZMAT^1*ID3 +b6*ZMAT^1*SIG+
 b7*ZMAT^2*SIGI +b8*ZMAT^2*ID3 +b9*ZMAT^2*SIG+
b10*ZMAT^3*SIGI+b11*ZMAT^3*ID3+b12*ZMAT^3*SIG+
b13*ZMAT^4*SIGI+b14*ZMAT^4*ID3+b15*ZMAT^4*SIG+
b16*ZMAT^5*SIGI+b17*ZMAT^5*ID3+b18*ZMAT^5*SIG;

%% Here are AA, BB and CC, as matrices with entries in \Q(z):
 
%% AA:=(1/5)*mat(
%% (6*z^5+2*z^4+4*z^3+4*z^2-z,-2*z^5-4*z^4+2*z^3-4*z,-z^5-z^4+3*z^3+2*z^2-4*z-3),
%% (-2*z^5-4*z^4-4*z^3-8*z^2-2*z-4,-4*z^5+2*z^3-5*z^2-2*z-4,-2*z^5-2*z^4-4*z^3-6*z^2-6*z-2),
%% (5*z^5-z^4+2*z^3+2*z^2+2*z+5,4*z^5-4*z^4+2*z^3+2*z^2,-2*z^5-7*z^4-6*z^3-4*z^2-2*z-6))$

%% BB:=(1/7)*mat(
%% (6*z^5-3*z^4-6*z^3-3*z^2-z,-2*z^5-7*z^4-z^3-5*z^2-5*z-1,7*z^5+5*z^4+z^3+2*z^2+8*z+5),
%% (z^5+6*z^4+8*z^3+7*z^2+3*z+3,6*z^5+3*z^4+12*z^3+5*z^2+3*z+6,z^5-4*z^4-z^3-4*z^2-6*z),
%% (1/2*(5*z^5-z^4+3*z^3-4*z^2-z+12),-8*z^5-z^4-7*z^3-5*z^2-2*z-5,-12*z^5-7*z^4-6*z^3-9*z^2-9*z-6))$

% Here are the inverses of AA and BB: 
AI:=(1/5)*mat(
(5*z^5+5*z^4+3*z^3+7*z^2+z+1,2*z^5+2*z^3-2*z+2,2*z^5+5*z^4+3*z^3+2*z^2+z+2),
(2*z^3+4*z,-3*z^5+4*z^4+2*z^3-2*z^2+2*z-2,-2*z^5-2*z^4-4*z^2-2*z),
(-z^5-3*z^4-z^3-z+2,-2*z^5-2*z^4-2*z^3+2*z^2-2*z-2,-2*z^5-4*z^4-5*z^3+2*z-4))$

BI:=(1/7)*mat(
(-2*z^5-5*z^4-2*z^3+7*z^2+z+1,-3*z^5+5*z^4+3*z^3-2*z^2+4*z+7,1/2*(11*z^5-3*z^4-7*z^3+6*z^2-6*z-15)),
(7*z^5+4*z^4+5*z^3-4*z^2-2*z-3,2*z^5+9*z^4+3*z^2-3*z+3,-3*z^5-5*z^4-6*z^3+z^2+2*z+4),
(z^5+7*z^4-3*z^3+6*z^2-z+4,-5*z^5-9*z^4+2*z^3-7*z^2-z-8,3*z^4+2*z^3-3*z^2+9*z+3))$

AASTAR:=sub(z=z^6,tp(AA))$
BBSTAR:=sub(z=z^6,tp(BB))$

% Checking that the elements $\xi$ satisfy $\iota(\xi)\xi=1$:
% The following are zero:
write "The following are zero, checking that AA and BB are unitary with respect to FF:";

AASTAR*FF*AA-FF;
BBSTAR*FF*BB-FF;

% Therefore, the inverses of AA and BB can be calculated using
AAI:=FFI*AASTAR*FF$
BBI:=FFI*BBSTAR*FF$
% The following are zero:
AAI-AI;
BBI-BI;

avec:=mat((a1),(a2),(a3),(a4),(a5),(a6),(a7),(a8),(a9),(a10),(a11),(a12),(a13),(a14),(a15),(a16),(a17),(a18))$
bvec:=mat((b1),(b2),(b3),(b4),(b5),(b6),(b7),(b8),(b9),(b10),(b11),(b12),(b13),(b14),(b15),(b16),(b17),(b18))$

% Checking the integality conditions: The following are zero:

CondMtxDM3Type1*avec-(1/5)*mat((-2),(0),(0),(-6),(-1),(-4),(-2),(4),(0),(2),(4),(2),(0),(2),(-4),(-2),(6),(-2));
CondMtxDM5Type2*avec-mat((-12),(-42),(-15),(-50),(-14),(-6),(-10),(10),(-12),(-22),(-15),(-14),(3),(-38),(-18),(-9),(-32),(-27));
CondMtxDM7Type1*avec-(1/5)*mat((103),(-252),(233),(-90),(62),(240),(-24),(216),(137),(-224),(208),(256),(-133),(72),(47),(41),(-294),(-19));

CondMtxDM3Type1*bvec-(1/7)*mat((2),(0),(-1),(6),(-1),(-5),(5),(-3),(-5),(-1),(-6),(-1),(2),(-3),(-7),(7),(6),(-2));
CondMtxDM5Type2*bvec-(1/7)*mat((51),(-285),(30),(162),(-170),(139),(-127),(-101),(-486),(132),(-318),(-186),(-130),(10),(-160),(-15),(-183),(-179));
CondMtxDM7Type1*bvec-mat((40),(61),(-16),(67),(-63),(-15),(-52),(-65),(-33),(-71),(8),(-45),(-34),(-78),(-3),(-44),(50),(-12));

% Relations: The following are zero:

BI*AI^3*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA^2*BB*AI*BB^2*AA*BB*AA*BB*AI^2*BB^2*AA*BB^3*AA*BI - gendt^6*ID3;
BB*AA*BB*AI*BB^2*AA*BB*AA*BB*AI^2*BB^2*AA*BB^2*AI*BI^2*AA*BI*AI^2*BI^3*AI*BI^2*AA^4*BB*AI*BB^2*AA*BB - gendt^2*ID3;
AI*BI^2*AA*BI*AI*BI^2*AI*BI^2*AA*BI*AI*BI^2*AI*BI^2*AA*BI*AI*BI*AI^3*BB^2*AA*BB^3*AA^2*BB*AI*BB^2*AA*BB*AA*BB*AA*BI*AI*BI - gendti^2*ID3;
AI*BI^3*AI*BI^2*AA^2*BI*AI*BI*AI*BI*AA*BB*AI*BB^2*AA*BB^2*AI^3*BB^2*AA*BB^3*AA^2*BB*AI*BB^2*AA*BB*AA*BI^2*AA^2*BB^2 - gendt^2*ID3;
AI*BB^2*AA*BB^3*AA*BI^2*AA^2*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BI*AI*BI^2*AI^3*BB^2*AA*BB^3*AA^2*BB*AI*BB^2*AA*BB*AA*BB - gendt^2*ID3;
AA*BI^3*AI*BI^2*AA^2*BB^2*AI*BI^3*AI*BI^2*AA*BI^3*AI*BI^2*AA^2*BI*AI*BI*AI*BI^2*AA*BI*AI*BI*AA*BB*AI*BI*AI^2*BI^3*AI*BI^2*AA - gendti^8*ID3;
BI*AA*BB*AI^2*BB^2*AA*BB^3*AA*BI^2*AI*BB^2*AI*BI^3*AI*BI^2*AA^2*BB*AI^2*BB^2*AA*BB^3*AA*BI^2*AI*BI^3*AI*BI^2*AA^3*BB^2*AI*BI - ID3;
AA*BI^2*AA^2*BB^2*AI*BI^3*AI*BI^2*AA^2*BI^2*AI*BB*AI*BI^3*AI*BI^2*AA^2*BI^3*AI*BI^2*AI*BB^2*AA*BB^3*AA^2*BB*AI*BB^2*AA*BB*AA*BB - gendti^2*ID3;
AA*BI*AI^3*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA*BI*AA*BB*AA*BB^2*AI*BI^3*AI*BI^2*AA^2*BI^2*AI*BI*AI*BI^2*AA*BI*AI*BI^2*AI*BI^2*AA*BI*AI*BI - gendti^2*ID3;
BI^2*AI*BI^2*AA*BI*AI*BB*AI*BI^3*AI*BI^2*AA^2*BI^3*AI^3*BB^2*AI^2*BB^2*AI^2*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA*BI^4*AI*BI^2*AA*BI*AI*BI - gendti^2*ID3;
BB^2*AI*BI*AI*BI*AI*BI^2*AA*BI*AI^2*BI^3*AI*BI^2*AA^3*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AI*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA*BI*AA*BB*AI - gendt^4*ID3;
AI^2*BB^2*AA*BB^3*AA^2*BB*AI*BB^2*AA^2*BB*AI*BB^2*AA*BB^2*AA*BB*AI*BB^2*AA*BB*AA*BB*AI*BI*AI*BI*AA*BB*AI*BB^2*AA*BB^2*AA*BB*AI*BB^2*AA*BB*AA*BB*AI*BI*AI*BI*AI - gendt^8*ID3;
BI^2*AI*BI^2*AA^2*BI^2*AA*BI*AI*BB*AA*BB*AA*BI*AI*BB*AI*BI^3*AI*BI^2*AA^2*BI^3*AI*BI^2*AA^2*BB*AA*BB*AI*BB^2*AA*BB*AA*BB^2*AI^2*BB^2*AA*BB^3*AA*BI^2*AI*BI - gendti^2*ID3;
AA^2*BB^3*AI^2*BB^2*AA*BB^3*AA*BI*AA*BB*AI*BB^2*AA*BB^2*AI*BI^2*AA*BB*AI^2*BB^2*AA*BB^3*AA*BI^2*AI^5*BB^2*AA*BB^3*AA^2*BB*AI*BB^2*AA*BB*AA*BB - gendt^8*ID3;
BI*AA^2*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BI*AI*BI*AA*BB^2*AI^2*BB^2*AA*BB^3*AA*BI^2*AI^2*BB^4*AI*BI^3*AI*BI^2*AA^2*BI^2*AI*BB*AA*BB*AI*BB^2*AA*BB*AA - ID3;
AA*BB*AI^2*BB^2*AA*BB^3*AA*BI^2*AI^2*BB^2*AI^2*BB^2*AA*BB^3*AA*BI^2*AA*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BB*AI*BI^2*AA*BI*AI^2*BI^3*AI*BI^2*AA^3*BB*AA*BB - ID3;
BI*AA*BI*AI^2*BI^3*AI*BI^2*AA^3*BI^2*AI^3*BB^2*AA*BB^3*AA*BB^2*AI*BI^3*AI*BI^2*AA^2*BI^2*AI*BB*AA*BB*AI*BB^2*AA*BB^2*AA*BB*AI*BB^2*AA*BB*AA*BB*AI*BI*AI*BI - ID3;
AI^2*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA*BI*AA*BB*AI*BB^2*AA*BB^2*AI*BI*AI*BI^2*AA*BI*AI^2*BI^3*AI*BI^2*AA^4*BB^2*AI^2*BB^2*AA*BB^3*AA*BI^2*AI^2*BB^2 - gendt^4*ID3;
AA*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BB^2*AI^2*BB^2*AA*BB^3*AA^2*BB*AI*BB*AI*BI^2*AA*BI*AI^2*BI^3*AI*BI^2*AA^3*BB*AI*BI^3*AI*BI^2*AA^2*BI^3*AI*BI^2*AA^3 - gendti^4*ID3;
BB*AI*BI^3*AI*BI^2*AA^2*BI*AI*BI*AI*BI^2*AA*BI*AI*BI*AA*BB*AI^4*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA*BI*AA*BB*AI*BI*AI*BI*AI^3*BB^2*AA*BB^3*AA^2*BB*AI*BB^2*AA*BI*AA - gendt^2*ID3;
BI*AI*BI*AA*BB^2*AI^2*BB^2*AA*BB^3*AA*BI^2*AI^2*BB^2*AI*BI*AI*BB*AA*BB*AI*BB^2*AA*BB*AA*BB*AI^2*BB^2*AA*BB^3*AA*BI*AI^3*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA*BI*AA*BB*AI - gendt^8*ID3;
AI*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA*BI*AA*BB*AI*BI*AI*BI^2*AA^2*BB^2*AI*BI^3*AI*BI^2*AA^2*BI^2*AI^2*BI^3*AI*BI^2*AA^3*BB*AA*BB*AI*BB^2*AA*BB^2*AI*BI^2*AA*BI*AI^2 - ID3;
AI^3*BB^2*AA*BB^3*AA^2*BB*AI*BB^2*AA*BB*AA*BI^2*AI*BI^2*AA*BI*AI*BI*AI*BI^2*AA*BB*AI^2*BB^2*AA*BB^3*AA*BI^2*AI^2*BB^2*AA*BB*AI*BB*AA*BB*AI*BB^2*AA*BB^2*AA*BB*AI*BB^2*AA*BB - gendt^6*ID3;
AI*BB*AA*BB*AI*BB^2*AA*BB*AA*BB*AI*BI^3*AI*BI^2*AA^2*BI*AI*BI*AI*BI^2*AA*BI*AI*BI*AA*BI^2*AI*BI^2*AA^3*BB^2*AI*BI^2*AA*BB*AI*BI^3*AI*BI^2*AA^2*BI*AI*BI*AI*BI^2*AA*BI*AI*BI*AA*BB*AI - gendti^6*ID3;
BI*AI*BB*AI*BI^3*AI*BI^2*AA^3*BI^2*AI*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA^2*BB*AA*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BB^2*AA^2*BB*AI*BB^2*AA*BB^2*AA*BB*AI*BB^2*AA*BB*AA*BB*AI - gendt^4*ID3;
AI^2*BB^2*AA*BB^3*AA^2*BB*AI*BB^2*AA*BB*AA*BB*AA*BI^2*AA^2*BI^2*AA^2*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BB^2*AA*BB^2*AA*BB*AI*BB^2*AA*BB*AA*BB*AI*BI*AI^2*BI^3*AI*BI^2*AA^2*BI^3*AI*BI^2 - ID3;
BI^2*AI*BI^2*AA^3*BB^2*AI*BI^2*AA*BI*AI^2*BI^3*AI*BI^2*AA^2*BI^2*AA*BB*AI^2*BB^2*AA*BB^3*AA*BI^2*AI^2*BI^2*AA^2*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BB^2*AA*BB^2*AA*BB*AI*BB^2*AA*BB*AA - gendti^2*ID3;
AA^2*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BI^3*AI*BI^2*AA*BI*AI*BB*AI*BI^3*AI*BI^2*AA^3*BI^2*AI*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA^2*BB*AI^2*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA^2*BB - gendt^2*ID3;
AI^2*BB^2*AA*BB^3*AA*BI^2*AA*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BB*AA*BB*AI*BB^2*AA*BB^2*AA*BB*AI*BB^2*AA*BB*AA*BB*AI*BI*AI*BI^2*AA*BB*AI*BI^3*AI*BI^2*AA^2*BI*AI*BI*AI*BI^2*AA*BI*AI*BI*AA*BB*AI*BB^2 - ID3;
BI*AI^3*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA*BI*AA*BB*AA^2*BB^3*AI^2*BB^2*AA*BB^3*AA*BI*AA*BB*AI*BB^2*AA*BB^4*AI*BI^3*AI*BI^2*AA^3*BI^2*AA^2*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI - gendt^4*ID3;
AA*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BB^2*AA*BB*AA*BB*AI*BB^2*AA*BB^3*AA*BI*AI*BI*AI*BI^2*AA*BI*AI*BI^2*AI*BI^2*AA*BI*AI*BI*AA*BB*AI*BI*AI*BI*AI*BI^2*AA*BI*AI^2*BI^3*AI*BI^2*AA^4*BI*AI*BI^2*AA - gendti^6*ID3;
AI^2*BB^2*AA*BB^2*AI*BI*AI*BI^2*AA*BI*AI*BI^2*AI*BI^3*AI*BI^2*AI^3*BB^2*AA*BB^3*AA^2*BB*AI*BB^2*AA*BB*AA*BB*AA^2*BB^2*AA*BB*AI*BB^2*AA*BB*AA*BB*AA*BI^2*AA^2*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BB^2 - gendt^2*ID3;
AI^4*BB^2*AA*BB^3*AA^2*BB*AI*BB^2*AA*BB*AA*BB*AA*BI^2*AI*BB^2*AA*BB^3*AA^2*BB*AI*BB^2*AA*BB*AA*BB*AA*BB^2*AI*BI^3*AI*BI^2*AA^2*BI^2*AI*BI*AI*BI^2*AA*BI*AI*BI*AA*BB*AI^2*BB^2*AA*BB^3*AA*BI^2 - gendt^4*ID3;
AI*BI*AI^2*BI^3*AI*BI^2*AA^2*BI^3*AI*BI^2*AA*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA^2*BB*AI^3*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA*BI*AA*BB*AI*BI*AI*BI*AA*BB*AI*BB^2*AA*BB^2*AA*BB*AI*BB^2*AA*BB*AA*BB - gendt^6*ID3;
BB*AI^2*BB^2*AA*BB^3*AA*BI*AA*BB*AI*BB^2*AA*BB^2*AI^3*BB^2*AA*BB^3*AA^2*BB*AI*BI*AI*BI^2*AA^2*BI^3*AI*BI^2*AA^3*BB^3*AA*BI*AI*BI*AI*BI^2*AA*BI*AI*BI^2*AI*BI^2*AA*BI*AI*BI*AA*BB*AI^2*BB^2*AA*BB^3*AA - gendt^2*ID3;
AI*BB^2*AA*BB^3*AA^2*BB*AI*BB^2*AA*BB*AA*BB*AI*BB^2*AI*BI*AI*BI*AI*BI^2*AA*BI*AI^2*BI^3*AI*BI^2*AA^4*BB^2*AI*BI^3*AI*BI^2*AA*BI^3*AI*BI^2*AA*BI*AI*BB*AI*BI^3*AI*BI^2*AA^2*BI^3*AI*BI^2*AA^2*BI^2 - gendti^6*ID3;
BI^2*AA^2*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BB^2*AA*BB^2*AI*BI^2*AA*BB*AI^2*BB^2*AA*BB^3*AA*BI^2*AI^4*BB^2*AA*BB^3*AA*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BI^3*AI*BI^2*AA^2*BB*AI^2*BB^2*AA*BB^3*AA*BI^2 - ID3;
AA^2*BB^2*AI*BI^3*AI*BI^2*AA^2*BI^2*AI*BI*AI*BI^2*AA*BI*AI*BI*AA*BB*AI*BI*AI*BI*AI*BI^2*AA*BI*AI^2*BI^3*AI*BI^2*AA^3*BI*AI*BI^2*AA*BI*AI*BI*AA^2*BI*AI*BB*AA*BB*AI*BB^2*AA*BB*AA*BB*AI^2*BB^2*AA*BB^3*AA*BI*AI*BI*AI*BI^2*AA - gendti^6*ID3;
BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA^2*BB*AA^2*BI^2*AA^2*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BB^2*AI^2*BB^2*AA*BB^2*AI*BB*AA*BB*AI*BB^2*AA*BB*AA*BB*AI^2*BB^2*AA*BB^3*AA*BI*AI*BB*AI*BI^3*AI*BI^2*AA^2*BI^3*AI*BI^2*AA - gendt^4*ID3;
BB^2*AI*BI*AI*BI^3*AI*BI^2*AA^2*BI*AI*BI*AI*BI^2*AA*BI*AI*BI^4*AI^3*BB^2*AA*BB^3*AA*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BI*AI*BI*AI*BI^2*AA*BI*AI^2*BI^3*AI*BI^2*AA^5*BB*AI^2*BB^2*AA*BB^3*AA*BI^2*AI^2 - gendti^6*ID3;
BB*AI*BI^3*AI*BI^2*AA^2*BI^3*AI*BI^2*AA*BI*AI*BB*AA*BB*AI*BB^2*AA*BB*AA*BB*AI^2*BB^2*AA*BB^3*AA*BI*AA*BB*AI^2*BB^2*AA*BB^3*AA*BI^2*AI^2*BB^2*AI^3*BB^2*AA*BB^3*AA*BI^2*AA*BB*AI*BB^2*AA*BB^2*AA*BB*AI*BB^2*AA*BB*AA*BB*AI*BI*AI - gendt^6*ID3;
AI*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA^2*BB*AI*BI*AI*BI^2*AA*BI*AI*BB*AA*BB^3*AA*BB^2*AA*BB*AI*BB^2*AA*BB*AA*BB*AA*BI^2*AA^2*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BB^2*AI^2*BB^2*AA*BB*AI*BI*AI*BI^2*AA*BI*AI*BI^2*AI*BI^2*AA*BI*AI*BI*AA*BI - gendt^2*ID3;
AA^2*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BB^2*AI^2*BB^2*AA*BB^3*AA^3*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BB^2*AA*BI^2*AI*BI^2*AA*BI*AI*BI*AA^2*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BB^2*AA*BB*AA*BB*AI*BB^2*AA*BB^2*AI*BI^2*AA*BI*AI*BI^2 - gendti^2*ID3;
AA*BB*AI^2*BB^2*AA*BB^3*AA*BI^2*AI*BI*AI*BI*AI*BI^2*AA*BI*AI*BI^2*AI^3*BB^2*AA*BB^3*AI*BB^2*AI*BI*AI*BI*AI*BI^2*AA*BI*AI^2*BI^3*AI*BI^2*AA^4*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI^4*BB^2*AA*BB^3*AA^2*BB*AI*BB^2*AA*BB*AA*BB - ID3;
AI^2*BB^2*AI^2*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA*BI*AA*BB*AA*BI*AI*BI*AI*BI^2*AA*BI*AI*BI*AA*BB*AA*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BI^3*AI*BI^2*AA*BI*AI*BI*AI*BI^2*AA*BB*AI*BI^3*AI*BI^2*AA^2*BI*AI*BI*AI*BI^2*AA*BI*AI*BI*AA*BB*AI*BI*AI*BI^2 - gendti^6*ID3;
BB^2*AA*BB*AI*BB^2*AA*BB*AA*BB*AA*BB^2*AI*BI^3*AI*BI^2*AA^2*BI^2*AI*BB*AA*BB*AI*BB^2*AA*BB*AA*BB*AI*BI^3*AI*BI^2*AI*BB^2*AA*BB^3*AA*BI^2*AI^6*BB^2*AA*BB^3*AA^2*BB*AI*BB^2*AA*BB*AA*BB*AA*BI^3*AI*BI^2*AI*BI^3*AI*BI^2*AA^3 - gendt^2*ID3;
BB*AI*BI^3*AI*BI^2*AA^2*BI^3*AI*BI^2*AA^2*BB^2*AI^2*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA*BI*AA*BB*AA*BI^2*AA^3*BB^3*AI^2*BB^2*AA*BB^3*AA*BI*AA*BB*AI*BB^2*AA*BB^4*AA*BB*AA*BI*AI*BB*AI*BI^3*AI*BI^2*AA^2*BI^3*AI*BI^2*AA^2 - gendt^2*ID3;
AI*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA^2*BB*AI*BI*AI*BI^2*AA*BI*AI*BB*AI*BI^3*AI*BI^2*AA^2*BI^3*AA*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BB^2*AA*BI^2*AI*BI^2*AA*BI*AI*BB*AA*BB^3*AA*BB^3*AI^2*BB^2*AA
*BB^3*AA*BI*AA*BB*AI*BB^2*AA*BB^2*AI*BI^2*AA*BI - gendt^2*ID3;
AI*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA^2*BB*AA^2*BB^2*AI*BI^3*AI*BI^2*AA^3*BB*AI^2*BB^2*AA*BB^3*AA*BI^2*AI^5*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA*BI*AA*BB*AI*BI*AI*BI*AI*BI^2*AA*BB*AI*BI^3*AI*BI^2*AA^2*BI*AI*BI*AI*BI^2*AA*BI*AI*BI*AA*BB*AI*BI*AI*BI^2 - gendt^2*ID3;
BI^2*AA*BB*AI^2*BB^2*AA*BB^3*AA*BI^2*AI^2*BB^2*AI*BI*AI*BI^3*AI*BI^2*AA^2*BI*AI*BI*AI*BI^2*AA*BI*AI*BI*AA*BI*AI*BB*AI*BI^3*AI*BI^2*AA^2*BI^3*AI*BI^2*AA^5*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BI^3*AI^3*BB^2*AA*BB^3*AA*BB^2*AI*BI^3*AI*BI^2*AA^2*BI - gendti^8*ID3;
BI*AA*BB*AI*BI*AI*BI*AI*BI^2*AA*BI*AI^3*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA*BI*AA*BB*AI*BI*AI*BI^2*AA*BB^2*AI*BI^3*AI*BI^2*AA^2*BI*AI*BB*AA*BB*AI*BB^2*AA*BB^2*AA*BB*AI*BB^2*AA*BB*AA*BB^2*AA*BB*AI^2*BB^2*AA*BB^3*AA*BI^2*AI^5*BB^2*AA*BB^3*AI^2*BB^2*AA*BB^3*AA - gendt^8*ID3;


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generators for $\bar\Gamma$ in the case $(a=7,p=2,\{7\})$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Firstly, the sequences of coefficients:

b1:= 5/7$
b2:=-3/7$
b3:=-1/7$
b4:= 3/7$
b5:= 1/7$
b6:=-9/7$
b7:=-6/7$
b8:=-2/7$
b9:=-3/7$
b10:=-1/7$
b11:=-5/7$
b12:= 3/7$
b13:= 4/7$
b14:=-8/7$
b15:=-5/7$
b16:= 2/7$
b17:=-4/7$
b18:=-6/7$

z1:=0$
z2:=0$
z3:=0$
z4:=0$
z5:=1$
z6:=0$
z7:=0$
z8:=0$
z9:=0$
z10:=0$
z11:=0$
z12:=0$
z13:=0$
z14:=0$
z15:=0$
z16:=0$
z17:=0$
z18:=0$

% Here are the matrix forms of the elements of $\cD$ corresponding
% to the above sequences of coefficients:

write "Here is the matrix for the element BB of the division algebra:";
BB:=
 b1*ZMAT^0*SIGI +b2*ZMAT^0*ID3 +b3*ZMAT^0*SIG+
 b4*ZMAT^1*SIGI +b5*ZMAT^1*ID3 +b6*ZMAT^1*SIG+
 b7*ZMAT^2*SIGI +b8*ZMAT^2*ID3 +b9*ZMAT^2*SIG+
b10*ZMAT^3*SIGI+b11*ZMAT^3*ID3+b12*ZMAT^3*SIG+
b13*ZMAT^4*SIGI+b14*ZMAT^4*ID3+b15*ZMAT^4*SIG+
b16*ZMAT^5*SIGI+b17*ZMAT^5*ID3+b18*ZMAT^5*SIG;

write "Here is the matrix for the element ZZ of the division algebra:";
ZZ:=
 z1*ZMAT^0*SIGI +z2*ZMAT^0*ID3 +z3*ZMAT^0*SIG+
 z4*ZMAT^1*SIGI +z5*ZMAT^1*ID3 +z6*ZMAT^1*SIG+
 z7*ZMAT^2*SIGI +z8*ZMAT^2*ID3 +z9*ZMAT^2*SIG+
z10*ZMAT^3*SIGI+z11*ZMAT^3*ID3+z12*ZMAT^3*SIG+
z13*ZMAT^4*SIGI+z14*ZMAT^4*ID3+z15*ZMAT^4*SIG+
z16*ZMAT^5*SIGI+z17*ZMAT^5*ID3+z18*ZMAT^5*SIG;

% Of course, ZZ is simply ZMAT.
write "The following is zero, checking that ZZ equals ZMAT:";

ZZ-ZMAT;

ZZSTAR:=sub(z=z^6,tp(ZZ))$
BBSTAR:=sub(z=z^6,tp(BB))$

% Checking that the elements $\xi$ satisfy $\iota(\xi)\xi=1$:
% The following are zero:
write "The following are zero, checking that BB and ZZ are unitary with respect to FF:";

BBSTAR*FF*BB-FF;
ZZSTAR*FF*ZZ-FF;

% Therefore, the inverses of ZZ and BB can be calculated using
ZZI:=FFI*ZZSTAR*FF$
BBI:=FFI*BBSTAR*FF$

% Explicitly:

%% BB:=(1/7)*mat(
%% (-4*z^5-8*z^4-5*z^3-2*z^2+z-3,-6*z^5-5*z^4+3*z^3-3*z^2-9*z-1,-2*z^5+3*z^4+z^3-8*z^2-3*z+2),
%% (z^5-5*z^4+3*z^3+4*z^2+5*z+6,5*z^5+3*z^4+z^3+6*z^2-3*z+2,-3*z^5-6*z^4-9*z^3-12*z^2-8*z-4),
%% (1/2*(17*z^5-z^4+9*z^3+12*z^2+z+18),-3*z^5+z^4-2*z^3+2*z^2-8*z+3,-z^5+5*z^4+4*z^3-4*z^2+2*z+1))$

%% BBI:=(1/7)*mat(
%% (-3*z^5-6*z^4-9*z^3-5*z^2-z-4,6*z^5+5*z^4+4*z^3+3*z^2+9*z+1,1/2*(-3*z^5+z^4+5*z^3+2*z^2-8*z+3)),
%% (-z^5-2*z^4-3*z^3-4*z^2+2*z+1,9*z^5+4*z^4+6*z^3+8*z^2+3*z+5,-4*z^5-z^4+2*z^3+5*z^2+z-3),
%% (-5*z^5-3*z^4-8*z^3-6*z^2-4*z-9,3*z^5-z^4+2*z^3+5*z^2+z+4,-6*z^5+2*z^4+3*z^3-3*z^2-2*z-1))$

zvec:=mat((z1),(z2),(z3),(z4),(z5),(z6),(z7),(z8),(z9),(z10),(z11),(z12),(z13),(z14),(z15),(z16),(z17),(z18))$
bvec:=mat((b1),(b2),(b3),(b4),(b5),(b6),(b7),(b8),(b9),(b10),(b11),(b12),(b13),(b14),(b15),(b16),(b17),(b18))$

write "The following are zero, checking that BB and ZZ satisfy the integrality conditions:";
condmtxdm3type1*zvec - mat((0),(0),(0),(0),(1),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0));
condmtxdm5type1*zvec - mat((0),(0),(0),(0),(1),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0));
condmtxdm7type2*zvec - mat((1),(0),(48),(2),(0),(0),(0),(0),(1),(0),(1),(0),(1),(0),(44),(3),(46),(1));

condmtxdm3type1*bvec - (1/7)*mat((5),(-3),(-1),(3),(1),(-9),(-6),(-2),(-3),(-1),(-5),(3),(4),(-8),(-5),(2),(-4),(-6));
condmtxdm5type1*bvec - (1/7)*mat((5),(-3),(-1),(3),(1),(-9),(-6),(-2),(-3),(-1),(-5),(3),(4),(-8),(-5),(2),(-4),(-6));
condmtxdm7type2*bvec - mat((-156),(-61),(-121),(-136),(-131),(-200),(-159),(-82),(-116),(-102),(-30),(-135),(-77),(-161),(22),(-69),(-57),(-89));

% Relations. Write BI and ZI in place of BBI and ZZI:

BI:=BBI$ ZI:=ZZI$

BB^3 - gendt*ID3;
ZZ^7 - ID3;
(BB*ZI^2*BB*ZI)^3 - gendt^2*ID3;
BI*ZZ*BB*ZZ^2*BB*ZZ^2*BI*ZI*BB*ZZ^2*BI*ZZ - ID3;
BB*ZZ^2*BI*ZI*BB*ZI*BB*ZZ^2*BI*ZI*BB*ZI*BB*ZI^3 - gendt*ID3;
BB*ZZ^2*BB*ZZ*BB*ZI^2*BI*ZZ*BB*ZI*BB*ZI^2*BI*ZZ^2 - gendt*ID3;


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generators for $\bar\Gamma$ in the case $(a=7,p=2,\{3,7\})$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Firstly, the sequences of coefficients:

a1:=13/21$
a2:=2/21$
a3:=-11/21$
a4:=12/21$
a5:=-10/21$
a6:=20/21$
a7:=4/21$
a8:=-8/21$
a9:=9/21$
a10:=10/21$
a11:=-6/21$
a12:=-9/21$
a13:=9/21$
a14:=3/21$
a15:=8/21$
a16:=-13/21$
a17:=12/21$
a18:=18/21$

b1:=0/3$
b2:=-1/3$
b3:=1/3$
b4:=4/3$
b5:=1/3$
b6:=-5/3$
b7:=3/3$
b8:=-3/3$
b9:=-2/3$
b10:=1/3$
b11:=-4/3$
b12:=0/3$
b13:=6/3$
b14:=-4/3$
b15:=-4/3$
b16:=7/3$
b17:=1/3$
b18:=-4/3$

% Here are the matrix forms of the elements of $\cD$ corresponding
% to the above sequences of coefficients:

write "Here is the matrix for the element AA of the division algebra:"$
AA:=
 a1*ZMAT^0*SIGI +a2*ZMAT^0*ID3 +a3*ZMAT^0*SIG+
 a4*ZMAT^1*SIGI +a5*ZMAT^1*ID3 +a6*ZMAT^1*SIG+
 a7*ZMAT^2*SIGI +a8*ZMAT^2*ID3 +a9*ZMAT^2*SIG+
a10*ZMAT^3*SIGI+a11*ZMAT^3*ID3+a12*ZMAT^3*SIG+
a13*ZMAT^4*SIGI+a14*ZMAT^4*ID3+a15*ZMAT^4*SIG+
a16*ZMAT^5*SIGI+a17*ZMAT^5*ID3+a18*ZMAT^5*SIG$

write "Here is the matrix for the element BB of the division algebra:"$
BB:=
 b1*ZMAT^0*SIGI +b2*ZMAT^0*ID3 +b3*ZMAT^0*SIG+
 b4*ZMAT^1*SIGI +b5*ZMAT^1*ID3 +b6*ZMAT^1*SIG+
 b7*ZMAT^2*SIGI +b8*ZMAT^2*ID3 +b9*ZMAT^2*SIG+
b10*ZMAT^3*SIGI+b11*ZMAT^3*ID3+b12*ZMAT^3*SIG+
b13*ZMAT^4*SIGI+b14*ZMAT^4*ID3+b15*ZMAT^4*SIG+
b16*ZMAT^5*SIGI+b17*ZMAT^5*ID3+b18*ZMAT^5*SIG$

% Explicitly:
%% AA:=(1/21)*mat(
%% (12*z^5+3*z^4-6*z^3-8*z^2-10*z+2,18*z^5+8*z^4-9*z^3+9*z^2+20*z-11,-22*z^5-9*z^4-3*z^3-4*z^2-5*z+8),
%% (-10*z^5-6*z^4-23*z^3+2*z^2-z+3,6*z^5-2*z^4+18*z^3-4*z^2+9*z+8,9*z^5+18*z^4+27*z^3+29*z^2+17*z-2),
%% (-22*z^5-9*z^4-10*z^3-25*z^2-5*z-34,23*z^5+25*z^4+13*z^3+22*z^2+17*z+26,-18*z^5-22*z^4-12*z^3-9*z^2-20*z-10))$

%% BB:=(1/3)*mat(
%% (z^5-4*z^4-4*z^3-3*z^2+z-1,-4*z^5-4*z^4-2*z^2-5*z+1,6*z^5+9*z^4+5*z^3+4*z^2+7*z+4),
%% (-z^5+2*z^4+6*z^3+3*z^2+5*z-1,4*z^5+z^4+5*z^3+5*z^2+3,-2*z^4-4*z^3-5*z^2-4*z+1),
%% (1/2*(10*z^5+4*z^4+7*z^3+8*z^2+5*z+15),-6*z^5-3*z^4-7*z^3-z^2-4*z-7,-5*z^5-z^3-5*z^2-4*z-2))$

% Here is the inverse of AA:
AI:=(1/21)*mat(
(2*z^5+4*z^4+13*z^3+22*z^2+10*z+12,-11*z^5-15*z^4-12*z^3+5*z^2-20*z-24,-13*z^5-12*z^4-11*z^3-17*z^2-9*z-8),
(3*z^5-8*z^4-5*z^3+5*z^2-6*z+4,-13*z^5+9*z^4-11*z^3-3*z^2-9*z-1,12*z^5+17*z^4+z^3-8*z^2-3*z-12),
(15*z^5-5*z^4+24*z^3+4*z^2+12*z-1,5*z^5+10*z^4+8*z^3-z^2-3*z+9,11*z^5+8*z^4-2*z^3+2*z^2+20*z+10))$

% Here is the inverse of BB:
BI:=(1/3)*mat(
(-4*z^5-5*z^4-5*z^3-z-2,z^5+z^4+3*z^3+2*z^2+2*z+5,1/2*(9*z^5+6*z^4+2*z^3+10*z^2+7*z+1)),
(z^5+4*z^4+3*z^3+z-2,5*z^5+5*z^4+z^3+4*z^2+3,-3*z^5-z^4-2*z^3-z^2-2*z+2),
(z^5+z^4-2*z^3+2*z^2+2*z+3,-3*z^5-3*z^4-2*z^3-2*z^2+z-5,-z^5+3*z^4+4*z^3-z^2+4*z+2))$

AASTAR:=sub(z=z^6,tp(AA))$
BBSTAR:=sub(z=z^6,tp(BB))$

% Checking that the elements $\xi$ satisfy $\iota(\xi)\xi=1$:
% The following are zero:
write "The following are zero, checking that AA and BB are unitary with respect to FF:";

AASTAR*FF*AA-FF;
BBSTAR*FF*BB-FF;

% Therefore, the inverses of AA and BB can be calculated using
AAI:=FFI*AASTAR*FF$
BBI:=FFI*BBSTAR*FF$
% The following are zero:
AAI-AI;
BBI-BI;

avec:=mat((a1),(a2),(a3),(a4),(a5),(a6),(a7),(a8),(a9),(a10),(a11),(a12),(a13),(a14),(a15),(a16),(a17),(a18))$
bvec:=mat((b1),(b2),(b3),(b4),(b5),(b6),(b7),(b8),(b9),(b10),(b11),(b12),(b13),(b14),(b15),(b16),(b17),(b18))$

write "The following are zero, checking that AA and BB satisfy the integrality conditions:";
condmtxdm3type2*avec - (1/7)*mat((96),(106),(139),(64),(121),(156),(83),(181),(91),(183),(28),(133),(65),(65),(62),(69),(24),(121));
condmtxdm5type1*avec - (1/21)*mat((13),(2),(-11),(12),(-10),(20),(4),(-8),(9),(10),(-6),(-9),(9),(3),(8),(-13),(12),(18));
condmtxdm7type2*avec - (1/3)*mat((330),(529),(121),(171),(89),(221),(28),(302),(22),(175),(5),(153),(189),(71),(147),(208),(295),(245));

condmtxdm3type2*bvec - mat((10),(17),(19),(25),(-41),(20),(-2),(8),(-1),(12),(-27),(-8),(-12),(2),(-17),(-13),(25),(-9));
condmtxdm5type1*bvec - (1/3)*mat((0),(-1),(1),(4),(1),(-5),(3),(-3),(-2),(1),(-4),(0),(6),(-4),(-4),(7),(1),(-4));
condmtxdm7type2*bvec - (1/3)*mat((-12),(200),(-239),(-250),(-392),(-343),(-190),(54),(-19),(-192),(214),(-159),(-106),(-276),(551),(-206),(328),(-60));

% Relations: The following are zero:

(AI*BB*AA*BB^3)^3 - gendt^4*ID3;
(BB*AI*BI*AI*BB^2*AA*BI^2*AI^2*BB*AA)^3 - gendt*ID3;
AI*BB^2*AI*BB*AI*BI*AI^2*BB*AA*BB*AA*BB*AA*BI^2 - gendt*ID3;
AA*BB*AI*BI*AA^2*BB^2*AI*BI*AA*BI*AI*BI*AA^2*BB*AA - ID3;
BI^2*AI^2*BB*AI*BI*AI*BI^2*AA*BB^2*AI*BB*AI*BI*AI*BI - gendti*ID3;
BI*AI*BB^2*AI*BB*AA*BB*AI^2*BI*AI*BI^2*AA*BB^2*AI*BI*AI*BB^2*AI - gendt*ID3;
AA*BB^2*AI*BI*AI*BB^2*AI*BB*AI*BB*AI*BI^3*AI^2*BB*AA*BB*AI*BI - gendt*ID3;
AI*BI*AA^2*BB^2*AI*BI^2*AA^2*BB*AI*BI*AA^2*BB^3*AA*BI*AA^2*BB^2 - gendt*ID3;
AI*BI^3*AI*BI^2*AA*BB^2*AI*BI*AI^3*BI^3*AI^2*BB*AA*BI*AI^2*BB - gendti^2*ID3;
BI*AA*BI*AI*BI*AA^2*BB^2*AI*BI^2*AA^2*BB^2*AI*BI*AI*BB^2*AA*BB*AA*BI*AA - ID3;
AA*BB*AI*BI*AA^2*BB^3*AI^2*BB^2*AA*BB*AA*BI*AA*BI*AA*BI^2*AI^2*BI*AI^2*BI*AI*BB^2*AI*BB - gendt*ID3;
AA*BB^2*AI*BB*AI*BB*AI*BI*AI*BI^2*AA^2*BI*AI*BI*AA^2*BI*AA^2*BB^2*AI*BB*AI*BB*AI*BI*AI*BI^5*AI^2*BB*AA^2 - gendti*ID3;


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generators for $\bar\Gamma$ in the case $(a=7,p=2,\{5,7\})$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Firstly, the sequences of coefficients:

a1:=-2/5$
a2:= 0/5$
a3:= 0/5$
a4:=-6/5$
a5:=-1/5$
a6:=-4/5$
a7:=-2/5$
a8:= 4/5$
a9:= 0/5$
a10:= 2/5$
a11:= 4/5$
a12:= 2/5$
a13:= 0/5$
a14:= 2/5$
a15:=-4/5$
a16:=-2/5$
a17:= 6/5$
a18:=-2/5$

b1:=-29/35$
b2:= 30/35$
b3:= 10/35$
b4:= -2/35$
b5:=-17/35$
b6:= -8/35$
b7:= 11/35$
b8:=-22/35$
b9:= -5/35$
b10:=-11/35$
b11:=-27/35$
b12:= 19/35$
b13:= -5/35$
b14:=-11/35$
b15:= 22/35$
b16:= 36/35$
b17:=-23/35$
b18:=-24/35$

d1:=-4/7$
d2:= 8/7$
d3:=-2/7$
d4:=-8/7$
d5:= 2/7$
d6:= 3/7$
d7:=-5/7$
d8:= 3/7$
d9:= 1/7$
d10:=-9/7$
d11:=-3/7$
d12:=-8/7$
d13:=-6/7$
d14:=-2/7$
d15:= 4/7$
d16:=-3/7$
d17:=-1/7$
d18:= 2/7$

A:=(1/5)*mat(
(6*z^5+2*z^4+4*z^3+4*z^2-z,-2*z^5-4*z^4+2*z^3-4*z,-z^5-z^4+3*z^3+2*z^2-4*z-3),
(-2*z^5-4*z^4-4*z^3-8*z^2-2*z-4,-4*z^5+2*z^3-5*z^2-2*z-4,-2*z^5-2*z^4-4*z^3-6*z^2-6*z-2),
(5*z^5-z^4+2*z^3+2*z^2+2*z+5,4*z^5-4*z^4+2*z^3+2*z^2,-2*z^5-7*z^4-6*z^3-4*z^2-2*z-6))$

AINV:=(1/5)*mat(
(5*z^5+5*z^4+3*z^3+7*z^2+z+1,2*z^5+2*z^3-2*z+2,2*z^5+5*z^4+3*z^3+2*z^2+z+2),
(2*z^3+4*z,-3*z^5+4*z^4+2*z^3-2*z^2+2*z-2,-2*z^5-2*z^4-4*z^2-2*z),
(-z^5-3*z^4-z^3-z+2,-2*z^5-2*z^4-2*z^3+2*z^2-2*z-2,-2*z^5-4*z^4-5*z^3+2*z-4))$

B:=(1/35)*mat(
(-23*z^5-11*z^4-27*z^3-22*z^2-17*z+30,-24*z^5+22*z^4+19*z^3-5*z^2-8*z+10,48*z^5+33*z^4+11*z^3+24*z^2+37*z-6),
(11*z^5+22*z^4+47*z^3+9*z^2+6*z-18,27*z^5+5*z^4+4*z^3+10*z^2+16*z+57,-19*z^5-24*z^4-43*z^3-27*z^2+3*z-9),
(1/2*(40*z^5+31*z^4+8*z^3+83*z^2-17*z+30),-47*z^5-38*z^4-36*z^3-41*z^2-25*z-65,-4*z^5+6*z^4+23*z^3+12*z^2+z+53))$

BINV:=(1/35)*mat(
(-5*z^5-10*z^4+6*z^3-6*z^2+17*z+47,24*z^5+20*z^4+9*z^3+5*z^2+36*z+39,1/2*(-47*z^5-10*z^4+27*z^3-27*z^2-11*z+68)),
(-25*z^5-15*z^4-26*z^3+5*z^2+8*z+25,-6*z^5-12*z^4-11*z^3+11*z^2-16*z+41,-9*z^5-4*z^4+15*z^3+27*z^2+11*z+30),
(-27*z^5+9*z^4-32*z^3-10*z^2-2*z-1,26*z^5+31*z^4+z^3+34*z^2+11*z+51,11*z^5+22*z^4+5*z^3-5*z^2-z+52))$

D:=(1/7)*mat(
(-z^5-2*z^4-3*z^3+3*z^2+2*z+8,2*z^5+4*z^4-8*z^3+z^2+3*z-2,3*z^5-z^4-5*z^3-2*z^2-6*z-3),
(9*z^5+4*z^4+6*z^3+z^2+3*z+5,3*z^5+6*z^4+2*z^3+5*z^2+z+11,8*z^5+9*z^4+10*z^3+11*z^2+12*z+6),
(1/2*(-15*z^5+5*z^4+4*z^3-4*z^2+2*z-13),-6*z^5-5*z^4+3*z^3-3*z^2-2*z-1,-2*z^5+3*z^4+z^3-z^2+4*z+9))$

DINV:=(1/7)*mat(
(z^5-5*z^4-4*z^3-3*z^2-2*z+6,-9*z^5-11*z^4+z^3-8*z^2-10*z-5,1/2*(-6*z^5-19*z^4-11*z^3+4*z^2+5*z-8)),
(-2*z^5+3*z^4-6*z^3-z^2-10*z-5,4*z^5+z^4+5*z^3+2*z^2-z+10,-z^5-9*z^4-10*z^3-11*z^2-12*z-6),
(11*z^5+z^4+5*z^3+2*z^2-z+10,6*z^5+5*z^4+4*z^3-4*z^2+9*z+1,-5*z^5-3*z^4-z^3-6*z^2-4*z+5))$


% Here are the matrix forms of the elements of $\cD$ corresponding
% to the above sequences of coefficients:

write "Here is the matrix for the element AA of the division algebra:"$
AA:=
 a1*ZMAT^0*SIGI +a2*ZMAT^0*ID3 +a3*ZMAT^0*SIG+
 a4*ZMAT^1*SIGI +a5*ZMAT^1*ID3 +a6*ZMAT^1*SIG+
 a7*ZMAT^2*SIGI +a8*ZMAT^2*ID3 +a9*ZMAT^2*SIG+
a10*ZMAT^3*SIGI+a11*ZMAT^3*ID3+a12*ZMAT^3*SIG+
a13*ZMAT^4*SIGI+a14*ZMAT^4*ID3+a15*ZMAT^4*SIG+
a16*ZMAT^5*SIGI+a17*ZMAT^5*ID3+a18*ZMAT^5*SIG$

write "Here is the matrix for the element BB of the division algebra:"$
BB:=
 b1*ZMAT^0*SIGI +b2*ZMAT^0*ID3 +b3*ZMAT^0*SIG+
 b4*ZMAT^1*SIGI +b5*ZMAT^1*ID3 +b6*ZMAT^1*SIG+
 b7*ZMAT^2*SIGI +b8*ZMAT^2*ID3 +b9*ZMAT^2*SIG+
b10*ZMAT^3*SIGI+b11*ZMAT^3*ID3+b12*ZMAT^3*SIG+
b13*ZMAT^4*SIGI+b14*ZMAT^4*ID3+b15*ZMAT^4*SIG+
b16*ZMAT^5*SIGI+b17*ZMAT^5*ID3+b18*ZMAT^5*SIG$

write "Here is the matrix for the element DD of the division algedra:"$
DD:=
 d1*ZMAT^0*SIGI +d2*ZMAT^0*ID3 +d3*ZMAT^0*SIG+
 d4*ZMAT^1*SIGI +d5*ZMAT^1*ID3 +d6*ZMAT^1*SIG+
 d7*ZMAT^2*SIGI +d8*ZMAT^2*ID3 +d9*ZMAT^2*SIG+
d10*ZMAT^3*SIGI+d11*ZMAT^3*ID3+d12*ZMAT^3*SIG+
d13*ZMAT^4*SIGI+d14*ZMAT^4*ID3+d15*ZMAT^4*SIG+
d16*ZMAT^5*SIGI+d17*ZMAT^5*ID3+d18*ZMAT^5*SIG$


AASTAR:=sub(z=z^6,tp(AA))$
BBSTAR:=sub(z=z^6,tp(BB))$
DDSTAR:=sub(z=z^6,tp(DD))$

% Checking that the elements $\xi$ satisfy $\iota(\xi)\xi=1$:
% The following are zero:
write "The following are zero, checking that AA, BB and DD are unitary with respect to FF:";

AASTAR*FF*AA-FF;
BBSTAR*FF*BB-FF;
DDSTAR*FF*DD-FF;

% Therefore, the inverses of AA, BB and DD can be calculated using
AI:=FFI*AASTAR*FF$
BI:=FFI*BBSTAR*FF$
DI:=FFI*DDSTAR*FF$
% The following are zero:
AI-AINV;
BI-BINV;
DI-DINV;

avec:=mat((a1),(a2),(a3),(a4),(a5),(a6),(a7),(a8),(a9),(a10),(a11),(a12),(a13),(a14),(a15),(a16),(a17),(a18))$
bvec:=mat((b1),(b2),(b3),(b4),(b5),(b6),(b7),(b8),(b9),(b10),(b11),(b12),(b13),(b14),(b15),(b16),(b17),(b18))$
dvec:=mat((d1),(d2),(d3),(d4),(d5),(d6),(d7),(d8),(d9),(d10),(d11),(d12),(d13),(d14),(d15),(d16),(d17),(d18))$

write "The following are zero, checking that AA, BB and DD satisfy the integrality conditions:";

CondMtxDM3type1*avec - (1/5)*mat((-2),(0),(0),(-6),(-1),(-4),(-2),(4),(0),(2),(4),(2),(0),(2),(-4),(-2),(6),(-2));
CondMtxDM3type1*bvec - (1/35)*mat((-29),(30),(10),(-2),(-17),(-8),(11),(-22),(-5),(-11),(-27),(19),(-5),(-11),(22),(36),(-23),(-24));
CondMtxDM3type1*dvec - (1/7)*mat((-4),(8),(-2),(-8),(2),(3),(-5),(3),(1),(-9),(-3),(-8),(-6),(-2),(4),(-3),(-1),(2));

CondMtxDM5type2*avec - mat((-12),(-42),(-15),(-50),(-14),(-6),(-10),(10),(-12),(-22),(-15),(-14),(3),(-38),(-18),(-9),(-32),(-27));
CondMtxDM5type2*bvec - (1/7)*mat((-167),(-6),(-197),(-67),(-140),(-166),(-140),(-38),(-220),(-284),(-337),(-169),(-79),(-206),(-81),(-13),(-170),(85));
CondMtxDM5type2*dvec - (1/7)*mat((-295),(-246),(-503),(-570),(-413),(-352),(-469),(-438),(-361),(-395),(-440),(-223),(-215),(-410),(-325),(-435),(-173),(-302));

CondMtxDM7type2*avec - (1/5)*mat((103),(-252),(-216),(278),(62),(240),(-118),(60),(137),(-224),(-27),(-82),(-133),(72),(-162),(-355),(-294),(-19));
CondMtxDM7type2*bvec - (1/5)*mat((-272),(-467),(-246),(-292),(-238),(-315),(157),(5),(-133),(-84),(-7),(-307),(-208),(-28),(-162),(-280),(-254),(-114));
CondMtxDM7type2*dvec - mat((-116),(-176),(-56),(-98),(-62),(-93),(-95),(-121),(-153),(-158),(-149),(-138),(-69),(-124),(-147),(-57),(-92),(-71));

% Relations: The following are zero:

AI*DD*BI*DD^2*AI*DD*AI*DD*BI*AA*DI*BB*DI*BB*DI*AA*DI*AA*DI - ID3;
DI*AA*DI*BB*DD*BI*DD*AI*DD*BB*AA*BB*AI^2*BI*DI*AA*DI*BB*DI*AI - ID3;
DI*BB*DI*AI^2*BI*DI*AA*DI*BB*DI*AI*DD*BB*DI*AI^2*BI*DI*AA*DI*BB*DI - gendti^2*ID3;
DI*AA*DI*BB*DD*AI*DD*AI*DD*AA^3*BI*DD*AI*DD*BB*AI*DI*AA*DI*AA*DI^2*BB*DI - ID3;
AI*BI*DI*AA*DI*BB*DI^2*BB*DI*AA*DD*BI*DD*AI*DD*BB*AA^2*BI*AA^2*BI*DD*AI*DD*BB - ID3;
DI*AA*BI*DD*AI*DD*BB*AA*BI*AA*BB*AI^2*BI*DI*AA*DI*BB*DI*BB*AI^2*BI*DI*AA*DI*BB - gendti*ID3;
DI*AA*DI*AA*BI*DD*AI*DD*BB*AA*DI*AI*BB*AI^2*BI*DI*AA*DI*BB*DI*AI*BB*AA^2*BI*AI - gendti*ID3;
BB*AA^3*BI*DD*AI*DD*BB*AA^2*DI*BI*DD*AI*DD*BB*AI^2*BI*DI*AA*DI*BB*DI^2*BB*DI - ID3;
DD*AA^3*BI*DD*AI*DD*BB*AA^2*DI*AI*DD*AI*DD*BI*DD^3*BI*DD*AI*DD*BB*AA^2*DD*AI - gendt^3*ID3;
BB*DI*AA*DI*AA*DI^2*BB*DI*AA*DI*BB*DI*BB*DI*AA*DI*AA*DI^2*BB*DI*AA*DI*BB*DI*AA*DI*BB*DI - gendti^3*ID3;
DD*AI*DD*AA*BB*AI^2*BI*DI*AA*DI*BB*AI*DD*AI*DD*BI*DI*BB*DI*AA*DI*AA*DD*BI*DD*AI*DD*BB*AA - gendt*ID3;
AA*BI*DD*AI*DD*BB*AA*BI*AI*DD*BI*DI*BB*AA*BI*DI*AA*DI*BB*AI^3*BI*DD*BB*DI*AA*DI*AA*DI - gendti*ID3;
BI*DI*AA*DI*BB*DI^2*BB*DI*AA*DI*AA*DD*BI*DD*AI*DD*BB*AA^2*DD*AI^2*BI*DI*AA*DI*BB*AI^3 - gendti*ID3;
AI*DD^4*BI*DD*AI*DD*BB*AA^2*DD*AI*DD*BI*DI*BB*AA*BI*DI*AA*DI*BB*AI^3*DI*AA*DI*AA*BI*DD - gendt*ID3;
BI*DI*AA*DI*BB*AI^2*BI*DI*AA*DI*BB*AI^2*BI*DI*AA*DI*BB*AI^3*DI*AA*DI*AA*BI*DD*AI*DD*BB*AA - gendti^2*ID3;
BI*AI*DI*AA*DI*BB*DD*BI*DD^2*BI*DD*AI*DD*BB*AA^3*BI*DI*AA*DI*BB*DI*BI*AA*BI*DD*AI*DD*BB*AA - ID3;
BI*DD*AI*DD*AA*DI*AA*DD*BI*DD*AI*DD*BB*AA^2*DD*AI^2*DI*AI^2*BI*DI*AA*DI*BB*DI^2*AI*DD*BI*DD - ID3;
DI*AI^2*BI*DI*AA*DI*BB*DI^2*AA*BI*DD*AI*DD*BB*AA*BI^2*DI*AA*DI*BB*DI*BI*AA*BI*DD*AI*DD*BB*AA - gendti^2*ID3;
AA*BB*AI*BI*DI*AA*DI*BB*AI*BB*AI^2*BI*DI*AA*DI*BB*DI*AI*BB*DI*AA*DI*BB*DI*BB*DI*AA*DI*AA*DI^2*BB - gendti^2*ID3;
BB*AI*BI*DI*AA*DI*BB*AI*DD*AI^3*BI*DI*AA*DI*BB*DI^2*BB*DI*AA*BB*AI*BI*DI*AA*DI*BB*AI*DD*AI*DD - gendti*ID3;
AA^2*BI*DD*AI*DD*BB*AA*BI*DI*AA*DI*BB*DD*BI*DD*AI*DD*BB*AI*BI*DD^2*AI*DD*AI*DD*BI*DD*BI*DD*AI*DD*BI - gendt^2*ID3;
AA*DD*AI^2*BI*DI*AA*DI*BB*AI^3*BI*DI*AA*DD*BI*DD*AI*DD*BB*AA^2*BI*DD^2*AI*DD*AI*DD*BI*DD*BI*DD*AI*DD*BI - gendt*ID3;
DI*AI^2*BI*DI*AA*DI*BB*DI^2*AI*DD*BI*DD^3*BI*DD*AI*DD*BB*AA*BB*DI*AA*DI*BB*DI*BB*DI*AA*DI*AA*DI^2*BB*AA - gendti*ID3;
DI*AA*BI*DD*AI*DD^2*BB*DI*AA*DI*AI^2*BI*DI*AA*DI*BB*DI^3*BB*AI^2*BI*DI*AA*DI*BB*DI^2*BB*DI^2*BB*DI*AA - gendti^3*ID3;
DD*AI*DD*BI*DD^2*AI*DD^3*BI*DD*AI*DD*BB*AA^2*DD*AA^2*BI*DD*AI*DD*AI*DD*BI*DI*BB*DI*AA*DI*AA*BI*DD*AI*DD - gendt^3*ID3;
BI*DI*AA*DI*BB*AI*DD*AI*DD*BI*DD*BB*DI*AA*BI*DI*AA*DI*BB*DI*BI*AA^2*BI*DD*AI*DD*BB*AA*BI*DI*AA*DI*BB*DD*AI^3 - gendti*ID3;
AI^2*BI*DI*AA*DI*BB*DI^2*BB*DI*BB*DI*AA*DI*AA*BI*DD*AI*DD*BB*AA^2*BI*AI*BI*DD^2*AI*DD*AI*DD*BI*DD*BI*DD*AI*DD - ID3;
DI*AI^2*BI*DI*AA*DI*BB*DI^2*AI*BI*DD^2*AI*DD*AI*DD*BI*DD*BI*DD*AI*DD*BI*DI*AA*DI*BB*AI*DD*AI*DD*BI*DD*BB*DI*AA*BB - ID3;
BB*AI^2*DI*AI^2*BI*DI*AA*DI*BB*DI^3*AA*BI*DD*AI*DD*AA*DI*BB^2*DI*AA*DI*AI^2*BI*DI*AA*DI*BB*DI^3*BB*AA^2 - gendti^3*ID3;
BI*DI*AA*DI*BB*AI*DD*AI*DD*AA*BB*AI*BI*DI*AA*DI*AA*BI*DD*AI*DD*BB*AA*BI*AI*DD*BI*DD*AI*DD*BB*DI*AA*BI*DD*AI*DD*BB*AA - gendt*ID3;
BI*DI*AA*DI*BB*DI^2*AA*DI*BB*DI*BB*DI*AA*DI*AA*DI^2*BB*AA^3*DI*BI*AA*BI*DD*AI*DD*BB*AA*BI*AI*DI*AA*DI*BB^2*DI*AA - gendti^3*ID3;
AI*BI*DI*AA*DI*BB*AI*DD*AI*DD*AA*BB*AA*BI*DD*AI*DD*BB*AA^2*BI*DD*AI*DD*BB^2*AI*BI*DI*AA*DI*BB*AI*BB*AI^2*BI*DI*AA*DI*BB - gendt*ID3;
DI*AA*DI*BB^2*DI*AA*DI*AI^2*BI*DI*AA*DI*BB*DI^3*BB*AA*DI*AA*DI*BB*DD*BI*DD*AI*DD*BB^2*AI^2*BI*DI*AA*DI*BB*DI^2*BB - gendti^2*ID3;
DD*AI^2*BI*DI*AA*DI*BB*AI*BI*AI*DI*AA*DI*AA*BI*DD*AI*DD*BB*DD*AA*BB*AI^2*BI*DI*AA*DI*BB*DI^2*AA*DD*BI*DD*AI*DD*BB*AA^2*DD - ID3;
AI*DD*BI*DD*AI*DI*AA*DI*BB*DD*BI*DD*AI*DD^3*AI*DD*AI*DD*BI*DD*BI*DD*AI*DD*BI*DI*AA*DI*BB*DD*BI*DD*AI*DD*BB^2*AI^2*BI*DI*AA*DI*BB*DI - gendt^2*ID3;
AA*BI*AI*BI*DD^2*AI*DD*AI*DD*BI*DD*BI*DD*AI*DD*AA*DI*AI^2*BI*DI*AA*DI*BB*DI*AI*DD*AI*DD*BI^2*DD*AI*DD*AA*DD^2*BI*DD*AI*DD*BB*AA^2*DD - gendt^2*ID3;
AI*DD*BI*DD*BI*DD^2*BI*DD*AI*DD*BB*AA^5*DI*BI*DD*AI*DD*BB*AI*BI*DI*AA*DI*BB*AI*BI*DD*AI*DD*BB*AA*BI*DD*BI*DD*AI*DD*BB*AA^2*BI*AI - gendt^2*ID3;
AI^2*BI*DI*AA*DI*BB*DI*BI*DD^2*BI*DD*AI*DD*BB*AA^3*BI*DI*AA*DI*BB*AI^2*BI*DI*AA*DI*BB*AI^5*BI*DI*AA*DI*BB*DI^2*BB*DI*BB*DI*AA*DI - gendti^3*ID3;
AA*DD*AI^2*BI*DI*AA*DI*BB*AI^3*DI*BI*DI*AA*DI*BB*AI*BB*DI*AA*DI*BB*DI*BB*DI*AA*DI*AA*DI^3*AA*DI*BB*DD*BI*DD*AI*DD*BB^2*AI^2*BI*DI*AA*DI*BB*DI^2 - gendti^3*ID3;
BI^2*DD^2*AI*DD*AI*DD*BI*DD*BI*DD*AI*DD*AA*BI*DI*AA*DI*BB*AI^5*BI*DI*AA*DI*BB*DI^2*BB*DI*BB*DI*AA^4*BI*DD*AI*DD*BB*AA^2*DI*AA*BI*DD*AI*DD*BB*AA - ID3;

;end;