(*{{2,3,8},{4,9,16}}
 I only add comments where there is non-trivial deviation from the "commented" and "bettercommented" file.
*)

Dynamic[{j,i,g, mark}]
Clear[n,B,R,Tile,A,U,mark,T,g,x];

n=144;

L=Expand[Cyclotomic[2,x^9]Cyclotomic[3,x^(16)]Cyclotomic[8,x^(9)]];


B=Array[0&,Length[L]];
For[j=2, j≤Length[L], B[[j]]=Mod[L[[j,2]],n];
j++];
B⋃{}

Tile=Array[0&,1000000];
R=Array[0&,n];
A=Array[200&,{n/Length[B],n,Length[B]}];
U=Array[0&,n/Length[B]];
U[[1]]=B;
mark=Array[1&,n/Length[B]+1];
T=Array[{0}&,n/Length[B]];
g=1;

For[i=1, i>0,
If[Length[U[[i]]]==n,  Tile[[g]]=T[[i]]⋃{};g=g+1;T[[i]]=T[[i-1]];mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];i=i-1,
For[j=1, j≤n,k=100;
If[Not[MemberQ[U[[i]],Mod[j,n]]], k=0;
For[m=1, m≤Length[B],
If[U[[i]]⋂ Mod[(j-B[[m]])+B,n]=={}, k=k+1; A[[i,j,m]]=Mod[j-B[[m]],n]];
m++]];
R[[j]]=k;
j++];
min=Min[R]; 
If[(Not[min==0]),
t=Catch[Do[If[R[[ii]]==min, Throw[ii]],{ii,n}]];
If[mark[[i]]≤Length[B],
For[b=mark[[i]], b<=Length[B],mark[[i]]=b+1;z=0;
If[Not[A[[i,t,b]]==200], U[[i+1]]=U[[i]]⋃Mod[A[[i,t,b]]+B,n]; T[[i+1]]=T[[i]]⋃{A[[i,t,b]]}; b=Length[B]+1;z=1;i=i+1;];
b++];
If[z==0, mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];i=i-1];
, mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];T[[i]]=T[[i-1]];i=i-1];
, mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];T[[i]]=T[[i-1]];i=i-1];
];
];

Tile=Drop[Tile⋃{},1];
Length[Tile]
Tile;



For[j=1, j≤Length[Tile], P=Tile[[j]];
For[i=2, i<=Length[P], 
If[MemberQ[Tile,Mod[P-P[[i]],n]⋃{}], Tile[[j]]=0;i=Length[P]+1];
i++];
j++]; 
Tile=Drop[Tile⋃{},1];
Length[Tile]
Tile

(*{{2,3,8},{4,9,16}}*)

(* Normally we should select the 6 emerging tiles according to Fourier zero-sets. This is done at a later step and it turns out that they fall into 1 class. Here i pick a representative and run the fill-out procedue to find non-periodic complements. There are 324 such complements. *)

Dynamic[{j,i,g, mark}]
Clear[n,B,R,Tile,A,U,mark,T,g,x];

n=144;


B={0,34,40,46,48,58,88,96,106,118,130,136};


Tile=Array[0&,1000000];
R=Array[0&,n];
A=Array[200&,{n/Length[B],n,Length[B]}];
U=Array[0&,n/Length[B]];
U[[1]]=B;
mark=Array[1&,n/Length[B]+1];
T=Array[{0}&,n/Length[B]];
g=1;

For[i=1, i>0,
If[Length[U[[i]]]==n,  Tile[[g]]=T[[i]]⋃{};g=g+1;T[[i]]=T[[i-1]];mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];i=i-1,
For[j=1, j≤n,k=100;
If[Not[MemberQ[U[[i]],Mod[j,n]]], k=0;
For[m=1, m≤Length[B],
If[U[[i]]⋂ Mod[(j-B[[m]])+B,n]=={}, k=k+1; A[[i,j,m]]=Mod[j-B[[m]],n]];
m++]];
R[[j]]=k;
j++];
min=Min[R]; 
If[(Not[min==0]),
t=Catch[Do[If[R[[ii]]==min, Throw[ii]],{ii,n}]];
If[mark[[i]]≤Length[B],
For[b=mark[[i]], b<=Length[B],mark[[i]]=b+1;z=0;
If[Not[A[[i,t,b]]==200], U[[i+1]]=U[[i]]⋃Mod[A[[i,t,b]]+B,n]; T[[i+1]]=T[[i]]⋃{A[[i,t,b]]}; b=Length[B]+1;z=1;i=i+1;];
b++];
If[z==0, mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];i=i-1];
, mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];T[[i]]=T[[i-1]];i=i-1];
, mark[[i]]=1;A[[i]]=Array[200&,{n,Length[B]}];T[[i]]=T[[i-1]];i=i-1];
];
];

Tile=Drop[Tile⋃{},1];
Length[Tile]
Tile;

H=Array[{0}&,Length[Tile[[1]]]+1];
K=Tile[[3]];
For[j=1, j≤Length[K], 
H[[j+1]]=Mod[(K[[j]]+B)⋃ H[[j]],n];
j++];
H[[Length[K]+1]]⋃{} 

For[j=1, j≤Length[Tile], P=Tile[[j]];
For[i=2, i<=Length[P], 
If[MemberQ[Tile,Mod[P-P[[i]],n]⋃{}], Tile[[j]]=0;i=Length[P]+1];
i++];
j++]; 
Tile=Drop[Tile⋃{},1];
Length[Tile]
Tile



(*{{2,3,8},{4,9,16}}

Here we check whether all 324 tiles with all 6. This is just crosscheking, as the 6 tiles have the same Fourier zeros. 
*)

Dynamic[{t1,t2,dd}]

T1={{0,16,29,44,57,73,80,93,108,109,124,137},{0,16,31,44,59,75,80,95,108,111,124,139},{0,16,32,37,53,69,73,89,105,108,124,140},{0,16,32,39,55,71,75,91,107,108,124,140},{0,16,32,41,57,73,77,93,108,109,124,140},{0,16,32,43,59,75,79,95,108,111,124,140},{0,16,32,45,61,77,81,97,108,113,124,140},{0,16,32,47,63,79,83,99,108,115,124,140},{0,17,21,36,49,53,64,80,85,100,116,129},{0,17,28,44,53,64,80,81,97,108,117,133},{0,17,28,44,61,64,80,93,97,108,125,129},{0,17,32,45,61,76,81,97,108,112,125,140},{0,17,33,36,53,64,69,80,97,100,116,133},{0,17,33,44,61,76,80,97,108,112,125,141},{0,17,33,49,53,69,76,85,92,108,112,128},{0,19,23,36,51,55,64,80,87,100,116,131},{0,19,28,44,55,64,80,83,99,108,119,135},{0,19,28,44,63,64,80,95,99,108,127,131},{0,19,32,47,63,76,83,99,108,112,127,140},{0,19,35,36,55,64,71,80,99,100,116,135},{0,19,35,44,63,76,80,99,108,112,127,143},{0,19,35,51,55,71,76,87,92,108,112,128},{0,19,36,47,63,64,80,83,99,100,116,127},{0,19,36,55,64,80,83,99,100,116,119,135},{0,20,36,41,57,64,73,77,93,100,109,128},{0,20,36,43,59,64,75,79,95,100,111,128},{0,20,36,45,61,64,77,81,97,100,113,128},{0,20,36,47,63,64,79,83,99,100,115,128},{0,20,36,49,64,65,81,85,100,101,117,128},{0,20,36,51,64,67,83,87,100,103,119,128},{0,20,36,53,64,69,85,89,100,105,121,128},{0,20,36,55,64,71,87,91,100,107,123,128},{0,21,25,36,53,57,64,80,89,100,116,133},{0,21,28,32,53,57,64,85,89,108,121,140},{0,21,28,37,53,57,64,73,89,92,108,128},{0,21,28,37,57,64,73,92,101,108,128,137},{0,21,28,44,57,64,80,85,101,108,121,137},{0,21,28,44,64,65,80,97,101,108,129,133},{0,21,32,36,57,64,68,85,100,101,121,137},{0,21,32,49,65,76,85,101,108,112,129,140},{0,21,32,53,57,76,85,89,108,112,121,140},{0,21,36,37,57,64,73,80,100,101,116,137},{0,21,36,49,64,65,80,85,100,101,116,129},{0,21,36,57,64,80,85,100,101,116,121,137},{0,21,37,53,57,73,76,89,92,108,112,128},{0,21,37,57,73,76,92,101,108,112,128,137},{0,23,27,36,55,59,64,80,91,100,116,135},{0,23,28,32,55,59,64,87,91,108,123,140},{0,23,28,39,55,59,64,75,91,92,108,128},{0,23,28,39,59,64,75,92,103,108,128,139},{0,23,28,44,55,59,64,80,87,91,108,123},{0,23,28,44,59,64,80,87,103,108,123,139},{0,23,28,44,64,67,80,99,103,108,131,135},{0,23,32,36,59,64,68,87,100,103,123,139},{0,23,32,51,67,76,87,103,108,112,131,140},{0,23,32,55,59,76,87,91,108,112,123,140},{0,23,36,39,59,64,75,80,100,103,116,139},{0,23,36,51,64,67,80,87,100,103,116,131},{0,23,36,55,59,64,80,87,91,100,116,123},{0,23,36,59,64,80,87,100,103,116,123,139},{0,23,39,55,59,75,76,91,92,108,112,128},{0,23,39,59,75,76,92,103,108,112,128,139},{0,23,44,51,67,76,80,87,103,108,112,131},{0,23,44,55,59,76,80,87,91,108,112,123},{0,23,44,59,76,80,87,103,108,112,123,139},{0,23,44,67,76,80,99,103,108,112,131,135},{0,25,28,41,57,61,64,77,92,93,108,128},{0,25,28,41,61,64,77,92,105,108,128,141},{0,25,28,44,57,61,64,80,89,93,108,125},{0,25,28,44,61,64,80,89,105,108,125,141},{0,25,29,36,57,61,64,80,93,100,116,137},{0,25,29,36,61,64,80,100,105,116,137,141},{0,25,29,44,61,76,80,105,108,112,137,141},{0,25,32,36,61,64,68,89,100,105,125,141},{0,25,32,53,69,76,89,105,108,112,133,140},{0,25,32,57,61,76,89,93,108,112,125,140},{0,25,36,41,61,64,77,80,100,105,116,141},{0,25,36,53,64,69,80,89,100,105,116,133},{0,25,36,57,61,64,80,89,93,100,116,125},{0,25,36,61,64,80,89,100,105,116,125,141},{0,25,41,57,61,76,77,92,93,108,112,128},{0,25,41,61,76,77,92,105,108,112,128,141},{0,25,44,53,69,76,80,89,105,108,112,133},{0,25,44,57,61,76,80,89,93,108,112,125},{0,25,44,61,76,80,89,105,108,112,125,141},{0,25,44,69,76,80,101,105,108,112,133,137},{0,27,28,43,59,63,64,79,92,95,108,128},{0,27,28,43,63,64,79,92,107,108,128,143},{0,27,28,44,59,63,64,80,91,95,108,127},{0,27,28,44,63,64,80,91,107,108,127,143},{0,27,31,36,59,63,64,80,95,100,116,139},{0,27,31,36,63,64,80,100,107,116,139,143},{0,27,31,44,63,76,80,107,108,112,139,143},{0,27,32,36,63,64,68,91,100,107,127,143},{0,27,32,55,71,76,91,107,108,112,135,140},{0,27,32,59,63,76,91,95,108,112,127,140},{0,27,36,43,63,64,79,80,100,107,116,143},{0,27,36,55,64,71,80,91,100,107,116,135},{0,27,36,59,63,64,80,91,95,100,116,127},{0,27,36,63,64,80,91,100,107,116,127,143},{0,27,43,59,63,76,79,92,95,108,112,128},{0,27,43,63,76,79,92,107,108,112,128,143},{0,27,44,55,71,76,80,91,107,108,112,135},{0,27,44,59,63,76,80,91,95,108,112,127},{0,27,44,63,76,80,91,107,108,112,127,143},{0,27,44,71,76,80,103,107,108,112,135,139},{0,28,29,44,61,64,65,80,93,97,108,129},{0,28,29,45,61,64,65,81,92,97,108,128},{0,28,31,32,59,64,75,95,108,111,139,140},{0,28,31,35,63,64,67,92,99,108,128,143},{0,28,31,44,63,64,67,80,95,99,108,131},{0,28,31,47,63,64,67,83,92,99,108,128},{0,28,32,33,61,64,77,97,108,113,140,141},{0,28,32,35,63,64,79,99,108,115,140,143},{0,28,32,45,61,64,77,81,97,108,113,140},{0,28,32,47,63,64,79,83,99,108,115,140},{0,28,32,49,64,65,81,85,101,108,117,140},{0,28,32,51,64,67,83,87,103,108,119,140},{0,28,32,53,64,69,85,89,105,108,121,140},{0,28,32,55,64,71,87,91,107,108,123,140},{0,28,32,57,64,73,89,93,108,109,125,140},{0,28,32,59,64,75,91,95,108,111,127,140},{0,28,33,44,61,64,77,80,97,108,113,141},{0,28,33,44,64,65,69,80,97,101,108,133},{0,28,33,49,64,65,69,85,92,101,108,128},{0,28,33,61,64,77,92,97,108,113,128,141},{0,28,35,44,63,64,79,80,99,108,115,143},{0,28,35,44,64,67,71,80,99,103,108,135},{0,28,35,51,64,67,71,87,92,103,108,128},{0,28,35,63,64,79,92,99,108,115,128,143},{0,28,37,53,64,69,73,89,92,105,108,128},{0,28,37,64,69,73,92,101,105,108,128,137},{0,28,39,55,64,71,75,91,92,107,108,128},{0,28,39,64,71,75,92,103,107,108,128,139},{0,28,41,57,64,73,77,92,93,108,109,128},{0,28,41,64,73,77,92,105,108,109,128,141},{0,28,43,59,64,75,79,92,95,108,111,128},{0,28,43,64,75,79,92,107,108,111,128,143},{0,28,44,49,64,65,80,81,85,101,108,117},{0,28,44,51,64,67,80,83,87,103,108,119},{0,28,44,53,64,69,80,85,89,105,108,121},{0,28,44,55,64,71,80,87,91,107,108,123},{0,28,44,57,64,73,80,89,93,108,109,125},{0,28,44,59,64,75,80,91,95,108,111,127},{0,28,44,61,64,77,80,93,97,108,113,129},{0,28,44,63,64,79,80,95,99,108,115,131},{0,28,44,64,65,80,81,97,101,108,117,133},{0,28,44,64,67,80,83,99,103,108,119,135},{0,28,44,64,69,80,85,101,105,108,121,137},{0,28,44,64,71,80,87,103,107,108,123,139},{0,28,44,64,73,80,89,105,108,109,125,141},{0,28,44,64,75,80,91,107,108,111,127,143},{0,28,45,61,64,77,81,92,97,108,113,128},{0,28,47,63,64,79,83,92,99,108,115,128},{0,28,49,64,65,81,85,92,101,108,117,128},{0,28,51,64,67,83,87,92,103,108,119,128},{0,28,53,64,69,85,89,92,105,108,121,128},{0,28,55,64,71,87,91,92,107,108,123,128},{0,29,32,33,61,65,76,97,108,112,140,141},{0,29,32,36,57,64,68,73,93,100,109,137},{0,29,32,36,64,68,73,100,105,109,137,141},{0,29,32,45,61,65,76,81,97,108,112,140},{0,29,32,57,73,76,93,108,109,112,137,140},{0,29,33,36,61,64,65,80,97,100,116,141},{0,29,33,44,61,65,76,80,97,108,112,141},{0,29,33,61,65,76,92,97,108,112,128,141},{0,29,36,45,61,64,65,80,81,97,100,116},{0,29,36,57,64,73,80,93,100,109,116,137},{0,29,36,61,64,65,80,93,97,100,116,129},{0,29,36,64,73,80,100,105,109,116,137,141},{0,29,44,57,73,76,80,93,108,109,112,137},{0,29,44,61,65,76,80,93,97,108,112,129},{0,29,44,73,76,80,105,108,109,112,137,141},{0,29,45,61,65,76,81,92,97,108,112,128},{0,29,57,73,76,92,93,108,109,112,128,137},{0,31,32,35,63,67,76,99,108,112,140,143},{0,31,32,36,59,64,68,75,95,100,111,139},{0,31,32,36,64,68,75,100,107,111,139,143},{0,31,32,47,63,67,76,83,99,108,112,140},{0,31,32,59,75,76,95,108,111,112,139,140},{0,31,35,36,63,64,67,80,99,100,116,143},{0,31,35,44,63,67,76,80,99,108,112,143},{0,31,35,63,67,76,92,99,108,112,128,143},{0,31,36,47,63,64,67,80,83,99,100,116},{0,31,36,59,64,75,80,95,100,111,116,139},{0,31,36,63,64,67,80,95,99,100,116,131},{0,31,36,64,75,80,100,107,111,116,139,143},{0,31,44,59,75,76,80,95,108,111,112,139},{0,31,44,63,67,76,80,95,99,108,112,131},{0,31,44,75,76,80,107,108,111,112,139,143},{0,31,47,63,67,76,83,92,99,108,112,128},{0,31,59,75,76,92,95,108,111,112,128,139},{0,32,33,36,61,64,68,77,97,100,113,141},{0,32,33,49,65,69,76,85,101,108,112,140},{0,32,33,61,76,77,97,108,112,113,140,141},{0,32,35,36,63,64,68,79,99,100,115,143},{0,32,35,51,67,71,76,87,103,108,112,140},{0,32,35,63,76,79,99,108,112,115,140,143},{0,32,36,41,64,68,73,77,100,105,109,141},{0,32,36,43,64,68,75,79,100,107,111,143},{0,32,36,45,61,64,68,77,81,97,100,113},{0,32,36,47,63,64,68,79,83,99,100,115},{0,32,36,49,64,65,68,81,85,100,101,117},{0,32,36,51,64,67,68,83,87,100,103,119},{0,32,36,53,64,68,69,85,89,100,105,121},{0,32,36,55,64,68,71,87,91,100,107,123},{0,32,36,57,64,68,73,89,93,100,109,125},{0,32,36,59,64,68,75,91,95,100,111,127},{0,32,36,61,64,68,77,93,97,100,113,129},{0,32,36,63,64,68,79,95,99,100,115,131},{0,32,36,64,65,68,81,97,100,101,117,133},{0,32,36,64,67,68,83,99,100,103,119,135},{0,32,36,64,68,69,85,100,101,105,121,137},{0,32,36,64,68,71,87,100,103,107,123,139},{0,32,36,64,68,73,89,100,105,109,125,141},{0,32,36,64,68,75,91,100,107,111,127,143},{0,32,37,53,69,73,76,89,105,108,112,140},{0,32,37,69,73,76,101,105,108,112,137,140},{0,32,39,55,71,75,76,91,107,108,112,140},{0,32,39,71,75,76,103,107,108,112,139,140},{0,32,41,57,73,76,77,93,108,109,112,140},{0,32,41,73,76,77,105,108,109,112,140,141},{0,32,43,59,75,76,79,95,108,111,112,140},{0,32,43,75,76,79,107,108,111,112,140,143},{0,32,45,61,76,77,81,97,108,112,113,140},{0,32,47,63,76,79,83,99,108,112,115,140},{0,32,49,65,76,81,85,101,108,112,117,140},{0,32,51,67,76,83,87,103,108,112,119,140},{0,32,53,69,76,85,89,105,108,112,121,140},{0,32,55,71,76,87,91,107,108,112,123,140},{0,32,57,73,76,89,93,108,109,112,125,140},{0,32,59,75,76,91,95,108,111,112,127,140},{0,32,61,76,77,93,97,108,112,113,129,140},{0,32,63,76,79,95,99,108,112,115,131,140},{0,33,36,49,64,65,69,80,85,100,101,116},{0,33,36,61,64,77,80,97,100,113,116,141},{0,33,36,64,65,69,80,97,100,101,116,133},{0,33,44,49,65,69,76,80,85,101,108,112},{0,33,44,61,76,77,80,97,108,112,113,141},{0,33,44,65,69,76,80,97,101,108,112,133},{0,33,49,65,69,76,85,92,101,108,112,128},{0,33,61,76,77,92,97,108,112,113,128,141},{0,33,65,69,76,92,97,101,108,112,128,133},{0,35,36,51,64,67,71,80,87,100,103,116},{0,35,36,63,64,79,80,99,100,115,116,143},{0,35,36,64,67,71,80,99,100,103,116,135},{0,35,44,51,67,71,76,80,87,103,108,112},{0,35,44,63,76,79,80,99,108,112,115,143},{0,35,44,67,71,76,80,99,103,108,112,135},{0,35,51,67,71,76,87,92,103,108,112,128},{0,35,63,76,79,92,99,108,112,115,128,143},{0,35,67,71,76,92,99,103,108,112,128,135},{0,36,37,53,64,69,73,80,89,100,105,116},{0,36,37,64,69,73,80,100,101,105,116,137},{0,36,39,55,64,71,75,80,91,100,107,116},{0,36,39,64,71,75,80,100,103,107,116,139},{0,36,41,57,64,73,77,80,93,100,109,116},{0,36,41,64,73,77,80,100,105,109,116,141},{0,36,43,59,64,75,79,80,95,100,111,116},{0,36,43,64,75,79,80,100,107,111,116,143},{0,36,45,61,64,77,80,81,97,100,113,116},{0,36,47,63,64,79,80,83,99,100,115,116},{0,36,49,64,65,80,81,85,100,101,116,117},{0,36,51,64,67,80,83,87,100,103,116,119},{0,36,53,64,69,80,85,89,100,105,116,121},{0,36,55,64,71,80,87,91,100,107,116,123},{0,36,57,64,73,80,89,93,100,109,116,125},{0,36,59,64,75,80,91,95,100,111,116,127},{0,36,61,64,77,80,93,97,100,113,116,129},{0,36,63,64,79,80,95,99,100,115,116,131},{0,36,64,65,80,81,97,100,101,116,117,133},{0,36,64,67,80,83,99,100,103,116,119,135},{0,36,64,69,80,85,100,101,105,116,121,137},{0,36,64,71,80,87,100,103,107,116,123,139},{0,36,64,73,80,89,100,105,109,116,125,141},{0,36,64,75,80,91,100,107,111,116,127,143},{0,37,44,53,69,73,76,80,89,105,108,112},{0,37,44,69,73,76,80,101,105,108,112,137},{0,37,53,69,73,76,89,92,105,108,112,128},{0,37,69,73,76,92,101,105,108,112,128,137},{0,39,44,55,71,75,76,80,91,107,108,112},{0,39,44,71,75,76,80,103,107,108,112,139},{0,39,55,71,75,76,91,92,107,108,112,128},{0,39,71,75,76,92,103,107,108,112,128,139},{0,41,44,57,73,76,77,80,93,108,109,112},{0,41,44,73,76,77,80,105,108,109,112,141},{0,41,57,73,76,77,92,93,108,109,112,128},{0,41,73,76,77,92,105,108,109,112,128,141},{0,43,44,59,75,76,79,80,95,108,111,112},{0,43,44,75,76,79,80,107,108,111,112,143},{0,43,59,75,76,79,92,95,108,111,112,128},{0,43,75,76,79,92,107,108,111,112,128,143},{0,44,45,61,76,77,80,81,97,108,112,113},{0,44,47,63,76,79,80,83,99,108,112,115},{0,44,49,65,76,80,81,85,101,108,112,117},{0,44,51,67,76,80,83,87,103,108,112,119},{0,44,53,69,76,80,85,89,105,108,112,121},{0,44,55,71,76,80,87,91,107,108,112,123},{0,44,57,73,76,80,89,93,108,109,112,125},{0,44,59,75,76,80,91,95,108,111,112,127},{0,44,61,76,77,80,93,97,108,112,113,129},{0,44,63,76,79,80,95,99,108,112,115,131},{0,44,65,76,80,81,97,101,108,112,117,133},{0,44,67,76,80,83,99,103,108,112,119,135},{0,44,69,76,80,85,101,105,108,112,121,137},{0,44,71,76,80,87,103,107,108,112,123,139},{0,44,73,76,80,89,105,108,109,112,125,141},{0,44,75,76,80,91,107,108,111,112,127,143},{0,45,61,76,77,81,92,97,108,112,113,128},{0,47,63,76,79,83,92,99,108,112,115,128},{0,49,65,76,81,85,92,101,108,112,117,128},{0,51,67,76,83,87,92,103,108,112,119,128},{0,53,69,76,85,89,92,105,108,112,121,128},{0,55,71,76,87,91,92,107,108,112,123,128},{0,57,73,76,89,92,93,108,109,112,125,128},{0,59,75,76,91,92,95,108,111,112,127,128},{0,61,76,77,92,93,97,108,112,113,128,129},{0,63,76,79,92,95,99,108,112,115,128,131},{0,65,76,81,92,97,101,108,112,117,128,133},{0,67,76,83,92,99,103,108,112,119,128,135},{0,69,76,85,92,101,105,108,112,121,128,137},{0,71,76,87,92,103,107,108,112,123,128,139},{0,73,76,89,92,105,108,109,112,125,128,141},{0,75,76,91,92,107,108,111,112,127,128,143}};

T2={{0,34,40,46,48,58,88,96,106,118,130,136},{0,34,42,48,60,72,82,90,120,130,132,138},{0,38,40,48,50,62,88,96,110,122,134,136},{0,38,46,48,60,72,86,94,120,132,134,142},{0,40,42,48,54,66,88,96,114,126,136,138},{0,40,46,48,58,70,88,96,118,130,136,142}};

NN={0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143};

H=Array[{0}&,13];
K=T1[[1]];
dd=1;
For[t1=1, t1≤Length[T1], K=T1[[t1]];
For[t2=1,t2≤Length[T2],B=T2[[t2]];H[[1]]={0};
For[j=1, j≤12, 
H[[j+1]]=Mod[(K[[j]]+B)⋃ H[[j]],n];
j++];
If[Not[H[[13]]⋃{}  == NN], dd=dd+1];
t2++];
t1++]; 
dd

(* And here, better late thatn never, we select the 324 and 6 tiles according to Fourier zero sets. The 324 fall into 2 classes, the 6 fall into 1. All tile with all. *)

T2={{0,16,29,44,57,73,80,93,108,109,124,137},{0,16,31,44,59,75,80,95,108,111,124,139},{0,16,32,37,53,69,73,89,105,108,124,140},{0,16,32,39,55,71,75,91,107,108,124,140},{0,16,32,41,57,73,77,93,108,109,124,140},{0,16,32,43,59,75,79,95,108,111,124,140},{0,16,32,45,61,77,81,97,108,113,124,140},{0,16,32,47,63,79,83,99,108,115,124,140},{0,17,21,36,49,53,64,80,85,100,116,129},{0,17,28,44,53,64,80,81,97,108,117,133},{0,17,28,44,61,64,80,93,97,108,125,129},{0,17,32,45,61,76,81,97,108,112,125,140},{0,17,33,36,53,64,69,80,97,100,116,133},{0,17,33,44,61,76,80,97,108,112,125,141},{0,17,33,49,53,69,76,85,92,108,112,128},{0,19,23,36,51,55,64,80,87,100,116,131},{0,19,28,44,55,64,80,83,99,108,119,135},{0,19,28,44,63,64,80,95,99,108,127,131},{0,19,32,47,63,76,83,99,108,112,127,140},{0,19,35,36,55,64,71,80,99,100,116,135},{0,19,35,44,63,76,80,99,108,112,127,143},{0,19,35,51,55,71,76,87,92,108,112,128},{0,19,36,47,63,64,80,83,99,100,116,127},{0,19,36,55,64,80,83,99,100,116,119,135},{0,20,36,41,57,64,73,77,93,100,109,128},{0,20,36,43,59,64,75,79,95,100,111,128},{0,20,36,45,61,64,77,81,97,100,113,128},{0,20,36,47,63,64,79,83,99,100,115,128},{0,20,36,49,64,65,81,85,100,101,117,128},{0,20,36,51,64,67,83,87,100,103,119,128},{0,20,36,53,64,69,85,89,100,105,121,128},{0,20,36,55,64,71,87,91,100,107,123,128},{0,21,25,36,53,57,64,80,89,100,116,133},{0,21,28,32,53,57,64,85,89,108,121,140},{0,21,28,37,53,57,64,73,89,92,108,128},{0,21,28,37,57,64,73,92,101,108,128,137},{0,21,28,44,57,64,80,85,101,108,121,137},{0,21,28,44,64,65,80,97,101,108,129,133},{0,21,32,36,57,64,68,85,100,101,121,137},{0,21,32,49,65,76,85,101,108,112,129,140},{0,21,32,53,57,76,85,89,108,112,121,140},{0,21,36,37,57,64,73,80,100,101,116,137},{0,21,36,49,64,65,80,85,100,101,116,129},{0,21,36,57,64,80,85,100,101,116,121,137},{0,21,37,53,57,73,76,89,92,108,112,128},{0,21,37,57,73,76,92,101,108,112,128,137},{0,23,27,36,55,59,64,80,91,100,116,135},{0,23,28,32,55,59,64,87,91,108,123,140},{0,23,28,39,55,59,64,75,91,92,108,128},{0,23,28,39,59,64,75,92,103,108,128,139},{0,23,28,44,55,59,64,80,87,91,108,123},{0,23,28,44,59,64,80,87,103,108,123,139},{0,23,28,44,64,67,80,99,103,108,131,135},{0,23,32,36,59,64,68,87,100,103,123,139},{0,23,32,51,67,76,87,103,108,112,131,140},{0,23,32,55,59,76,87,91,108,112,123,140},{0,23,36,39,59,64,75,80,100,103,116,139},{0,23,36,51,64,67,80,87,100,103,116,131},{0,23,36,55,59,64,80,87,91,100,116,123},{0,23,36,59,64,80,87,100,103,116,123,139},{0,23,39,55,59,75,76,91,92,108,112,128},{0,23,39,59,75,76,92,103,108,112,128,139},{0,23,44,51,67,76,80,87,103,108,112,131},{0,23,44,55,59,76,80,87,91,108,112,123},{0,23,44,59,76,80,87,103,108,112,123,139},{0,23,44,67,76,80,99,103,108,112,131,135},{0,25,28,41,57,61,64,77,92,93,108,128},{0,25,28,41,61,64,77,92,105,108,128,141},{0,25,28,44,57,61,64,80,89,93,108,125},{0,25,28,44,61,64,80,89,105,108,125,141},{0,25,29,36,57,61,64,80,93,100,116,137},{0,25,29,36,61,64,80,100,105,116,137,141},{0,25,29,44,61,76,80,105,108,112,137,141},{0,25,32,36,61,64,68,89,100,105,125,141},{0,25,32,53,69,76,89,105,108,112,133,140},{0,25,32,57,61,76,89,93,108,112,125,140},{0,25,36,41,61,64,77,80,100,105,116,141},{0,25,36,53,64,69,80,89,100,105,116,133},{0,25,36,57,61,64,80,89,93,100,116,125},{0,25,36,61,64,80,89,100,105,116,125,141},{0,25,41,57,61,76,77,92,93,108,112,128},{0,25,41,61,76,77,92,105,108,112,128,141},{0,25,44,53,69,76,80,89,105,108,112,133},{0,25,44,57,61,76,80,89,93,108,112,125},{0,25,44,61,76,80,89,105,108,112,125,141},{0,25,44,69,76,80,101,105,108,112,133,137},{0,27,28,43,59,63,64,79,92,95,108,128},{0,27,28,43,63,64,79,92,107,108,128,143},{0,27,28,44,59,63,64,80,91,95,108,127},{0,27,28,44,63,64,80,91,107,108,127,143},{0,27,31,36,59,63,64,80,95,100,116,139},{0,27,31,36,63,64,80,100,107,116,139,143},{0,27,31,44,63,76,80,107,108,112,139,143},{0,27,32,36,63,64,68,91,100,107,127,143},{0,27,32,55,71,76,91,107,108,112,135,140},{0,27,32,59,63,76,91,95,108,112,127,140},{0,27,36,43,63,64,79,80,100,107,116,143},{0,27,36,55,64,71,80,91,100,107,116,135},{0,27,36,59,63,64,80,91,95,100,116,127},{0,27,36,63,64,80,91,100,107,116,127,143},{0,27,43,59,63,76,79,92,95,108,112,128},{0,27,43,63,76,79,92,107,108,112,128,143},{0,27,44,55,71,76,80,91,107,108,112,135},{0,27,44,59,63,76,80,91,95,108,112,127},{0,27,44,63,76,80,91,107,108,112,127,143},{0,27,44,71,76,80,103,107,108,112,135,139},{0,28,29,44,61,64,65,80,93,97,108,129},{0,28,29,45,61,64,65,81,92,97,108,128},{0,28,31,32,59,64,75,95,108,111,139,140},{0,28,31,35,63,64,67,92,99,108,128,143},{0,28,31,44,63,64,67,80,95,99,108,131},{0,28,31,47,63,64,67,83,92,99,108,128},{0,28,32,33,61,64,77,97,108,113,140,141},{0,28,32,35,63,64,79,99,108,115,140,143},{0,28,32,45,61,64,77,81,97,108,113,140},{0,28,32,47,63,64,79,83,99,108,115,140},{0,28,32,49,64,65,81,85,101,108,117,140},{0,28,32,51,64,67,83,87,103,108,119,140},{0,28,32,53,64,69,85,89,105,108,121,140},{0,28,32,55,64,71,87,91,107,108,123,140},{0,28,32,57,64,73,89,93,108,109,125,140},{0,28,32,59,64,75,91,95,108,111,127,140},{0,28,33,44,61,64,77,80,97,108,113,141},{0,28,33,44,64,65,69,80,97,101,108,133},{0,28,33,49,64,65,69,85,92,101,108,128},{0,28,33,61,64,77,92,97,108,113,128,141},{0,28,35,44,63,64,79,80,99,108,115,143},{0,28,35,44,64,67,71,80,99,103,108,135},{0,28,35,51,64,67,71,87,92,103,108,128},{0,28,35,63,64,79,92,99,108,115,128,143},{0,28,37,53,64,69,73,89,92,105,108,128},{0,28,37,64,69,73,92,101,105,108,128,137},{0,28,39,55,64,71,75,91,92,107,108,128},{0,28,39,64,71,75,92,103,107,108,128,139},{0,28,41,57,64,73,77,92,93,108,109,128},{0,28,41,64,73,77,92,105,108,109,128,141},{0,28,43,59,64,75,79,92,95,108,111,128},{0,28,43,64,75,79,92,107,108,111,128,143},{0,28,44,49,64,65,80,81,85,101,108,117},{0,28,44,51,64,67,80,83,87,103,108,119},{0,28,44,53,64,69,80,85,89,105,108,121},{0,28,44,55,64,71,80,87,91,107,108,123},{0,28,44,57,64,73,80,89,93,108,109,125},{0,28,44,59,64,75,80,91,95,108,111,127},{0,28,44,61,64,77,80,93,97,108,113,129},{0,28,44,63,64,79,80,95,99,108,115,131},{0,28,44,64,65,80,81,97,101,108,117,133},{0,28,44,64,67,80,83,99,103,108,119,135},{0,28,44,64,69,80,85,101,105,108,121,137},{0,28,44,64,71,80,87,103,107,108,123,139},{0,28,44,64,73,80,89,105,108,109,125,141},{0,28,44,64,75,80,91,107,108,111,127,143},{0,28,45,61,64,77,81,92,97,108,113,128},{0,28,47,63,64,79,83,92,99,108,115,128},{0,28,49,64,65,81,85,92,101,108,117,128},{0,28,51,64,67,83,87,92,103,108,119,128},{0,28,53,64,69,85,89,92,105,108,121,128},{0,28,55,64,71,87,91,92,107,108,123,128},{0,29,32,33,61,65,76,97,108,112,140,141},{0,29,32,36,57,64,68,73,93,100,109,137},{0,29,32,36,64,68,73,100,105,109,137,141},{0,29,32,45,61,65,76,81,97,108,112,140},{0,29,32,57,73,76,93,108,109,112,137,140},{0,29,33,36,61,64,65,80,97,100,116,141},{0,29,33,44,61,65,76,80,97,108,112,141},{0,29,33,61,65,76,92,97,108,112,128,141},{0,29,36,45,61,64,65,80,81,97,100,116},{0,29,36,57,64,73,80,93,100,109,116,137},{0,29,36,61,64,65,80,93,97,100,116,129},{0,29,36,64,73,80,100,105,109,116,137,141},{0,29,44,57,73,76,80,93,108,109,112,137},{0,29,44,61,65,76,80,93,97,108,112,129},{0,29,44,73,76,80,105,108,109,112,137,141},{0,29,45,61,65,76,81,92,97,108,112,128},{0,29,57,73,76,92,93,108,109,112,128,137},{0,31,32,35,63,67,76,99,108,112,140,143},{0,31,32,36,59,64,68,75,95,100,111,139},{0,31,32,36,64,68,75,100,107,111,139,143},{0,31,32,47,63,67,76,83,99,108,112,140},{0,31,32,59,75,76,95,108,111,112,139,140},{0,31,35,36,63,64,67,80,99,100,116,143},{0,31,35,44,63,67,76,80,99,108,112,143},{0,31,35,63,67,76,92,99,108,112,128,143},{0,31,36,47,63,64,67,80,83,99,100,116},{0,31,36,59,64,75,80,95,100,111,116,139},{0,31,36,63,64,67,80,95,99,100,116,131},{0,31,36,64,75,80,100,107,111,116,139,143},{0,31,44,59,75,76,80,95,108,111,112,139},{0,31,44,63,67,76,80,95,99,108,112,131},{0,31,44,75,76,80,107,108,111,112,139,143},{0,31,47,63,67,76,83,92,99,108,112,128},{0,31,59,75,76,92,95,108,111,112,128,139},{0,32,33,36,61,64,68,77,97,100,113,141},{0,32,33,49,65,69,76,85,101,108,112,140},{0,32,33,61,76,77,97,108,112,113,140,141},{0,32,35,36,63,64,68,79,99,100,115,143},{0,32,35,51,67,71,76,87,103,108,112,140},{0,32,35,63,76,79,99,108,112,115,140,143},{0,32,36,41,64,68,73,77,100,105,109,141},{0,32,36,43,64,68,75,79,100,107,111,143},{0,32,36,45,61,64,68,77,81,97,100,113},{0,32,36,47,63,64,68,79,83,99,100,115},{0,32,36,49,64,65,68,81,85,100,101,117},{0,32,36,51,64,67,68,83,87,100,103,119},{0,32,36,53,64,68,69,85,89,100,105,121},{0,32,36,55,64,68,71,87,91,100,107,123},{0,32,36,57,64,68,73,89,93,100,109,125},{0,32,36,59,64,68,75,91,95,100,111,127},{0,32,36,61,64,68,77,93,97,100,113,129},{0,32,36,63,64,68,79,95,99,100,115,131},{0,32,36,64,65,68,81,97,100,101,117,133},{0,32,36,64,67,68,83,99,100,103,119,135},{0,32,36,64,68,69,85,100,101,105,121,137},{0,32,36,64,68,71,87,100,103,107,123,139},{0,32,36,64,68,73,89,100,105,109,125,141},{0,32,36,64,68,75,91,100,107,111,127,143},{0,32,37,53,69,73,76,89,105,108,112,140},{0,32,37,69,73,76,101,105,108,112,137,140},{0,32,39,55,71,75,76,91,107,108,112,140},{0,32,39,71,75,76,103,107,108,112,139,140},{0,32,41,57,73,76,77,93,108,109,112,140},{0,32,41,73,76,77,105,108,109,112,140,141},{0,32,43,59,75,76,79,95,108,111,112,140},{0,32,43,75,76,79,107,108,111,112,140,143},{0,32,45,61,76,77,81,97,108,112,113,140},{0,32,47,63,76,79,83,99,108,112,115,140},{0,32,49,65,76,81,85,101,108,112,117,140},{0,32,51,67,76,83,87,103,108,112,119,140},{0,32,53,69,76,85,89,105,108,112,121,140},{0,32,55,71,76,87,91,107,108,112,123,140},{0,32,57,73,76,89,93,108,109,112,125,140},{0,32,59,75,76,91,95,108,111,112,127,140},{0,32,61,76,77,93,97,108,112,113,129,140},{0,32,63,76,79,95,99,108,112,115,131,140},{0,33,36,49,64,65,69,80,85,100,101,116},{0,33,36,61,64,77,80,97,100,113,116,141},{0,33,36,64,65,69,80,97,100,101,116,133},{0,33,44,49,65,69,76,80,85,101,108,112},{0,33,44,61,76,77,80,97,108,112,113,141},{0,33,44,65,69,76,80,97,101,108,112,133},{0,33,49,65,69,76,85,92,101,108,112,128},{0,33,61,76,77,92,97,108,112,113,128,141},{0,33,65,69,76,92,97,101,108,112,128,133},{0,35,36,51,64,67,71,80,87,100,103,116},{0,35,36,63,64,79,80,99,100,115,116,143},{0,35,36,64,67,71,80,99,100,103,116,135},{0,35,44,51,67,71,76,80,87,103,108,112},{0,35,44,63,76,79,80,99,108,112,115,143},{0,35,44,67,71,76,80,99,103,108,112,135},{0,35,51,67,71,76,87,92,103,108,112,128},{0,35,63,76,79,92,99,108,112,115,128,143},{0,35,67,71,76,92,99,103,108,112,128,135},{0,36,37,53,64,69,73,80,89,100,105,116},{0,36,37,64,69,73,80,100,101,105,116,137},{0,36,39,55,64,71,75,80,91,100,107,116},{0,36,39,64,71,75,80,100,103,107,116,139},{0,36,41,57,64,73,77,80,93,100,109,116},{0,36,41,64,73,77,80,100,105,109,116,141},{0,36,43,59,64,75,79,80,95,100,111,116},{0,36,43,64,75,79,80,100,107,111,116,143},{0,36,45,61,64,77,80,81,97,100,113,116},{0,36,47,63,64,79,80,83,99,100,115,116},{0,36,49,64,65,80,81,85,100,101,116,117},{0,36,51,64,67,80,83,87,100,103,116,119},{0,36,53,64,69,80,85,89,100,105,116,121},{0,36,55,64,71,80,87,91,100,107,116,123},{0,36,57,64,73,80,89,93,100,109,116,125},{0,36,59,64,75,80,91,95,100,111,116,127},{0,36,61,64,77,80,93,97,100,113,116,129},{0,36,63,64,79,80,95,99,100,115,116,131},{0,36,64,65,80,81,97,100,101,116,117,133},{0,36,64,67,80,83,99,100,103,116,119,135},{0,36,64,69,80,85,100,101,105,116,121,137},{0,36,64,71,80,87,100,103,107,116,123,139},{0,36,64,73,80,89,100,105,109,116,125,141},{0,36,64,75,80,91,100,107,111,116,127,143},{0,37,44,53,69,73,76,80,89,105,108,112},{0,37,44,69,73,76,80,101,105,108,112,137},{0,37,53,69,73,76,89,92,105,108,112,128},{0,37,69,73,76,92,101,105,108,112,128,137},{0,39,44,55,71,75,76,80,91,107,108,112},{0,39,44,71,75,76,80,103,107,108,112,139},{0,39,55,71,75,76,91,92,107,108,112,128},{0,39,71,75,76,92,103,107,108,112,128,139},{0,41,44,57,73,76,77,80,93,108,109,112},{0,41,44,73,76,77,80,105,108,109,112,141},{0,41,57,73,76,77,92,93,108,109,112,128},{0,41,73,76,77,92,105,108,109,112,128,141},{0,43,44,59,75,76,79,80,95,108,111,112},{0,43,44,75,76,79,80,107,108,111,112,143},{0,43,59,75,76,79,92,95,108,111,112,128},{0,43,75,76,79,92,107,108,111,112,128,143},{0,44,45,61,76,77,80,81,97,108,112,113},{0,44,47,63,76,79,80,83,99,108,112,115},{0,44,49,65,76,80,81,85,101,108,112,117},{0,44,51,67,76,80,83,87,103,108,112,119},{0,44,53,69,76,80,85,89,105,108,112,121},{0,44,55,71,76,80,87,91,107,108,112,123},{0,44,57,73,76,80,89,93,108,109,112,125},{0,44,59,75,76,80,91,95,108,111,112,127},{0,44,61,76,77,80,93,97,108,112,113,129},{0,44,63,76,79,80,95,99,108,112,115,131},{0,44,65,76,80,81,97,101,108,112,117,133},{0,44,67,76,80,83,99,103,108,112,119,135},{0,44,69,76,80,85,101,105,108,112,121,137},{0,44,71,76,80,87,103,107,108,112,123,139},{0,44,73,76,80,89,105,108,109,112,125,141},{0,44,75,76,80,91,107,108,111,112,127,143},{0,45,61,76,77,81,92,97,108,112,113,128},{0,47,63,76,79,83,92,99,108,112,115,128},{0,49,65,76,81,85,92,101,108,112,117,128},{0,51,67,76,83,87,92,103,108,112,119,128},{0,53,69,76,85,89,92,105,108,112,121,128},{0,55,71,76,87,91,92,107,108,112,123,128},{0,57,73,76,89,92,93,108,109,112,125,128},{0,59,75,76,91,92,95,108,111,112,127,128},{0,61,76,77,92,93,97,108,112,113,128,129},{0,63,76,79,92,95,99,108,112,115,128,131},{0,65,76,81,92,97,101,108,112,117,128,133},{0,67,76,83,92,99,103,108,112,119,128,135},{0,69,76,85,92,101,105,108,112,121,128,137},{0,71,76,87,92,103,107,108,112,123,128,139},{0,73,76,89,92,105,108,109,112,125,128,141},{0,75,76,91,92,107,108,111,112,127,128,143}};


Clear[H];

Length[T2]

n=144;
H=Array[0&,{Length[T2],n}];
MF=Array[0&,100];
MT=Array[0&,100];

For[j=1, j≤Length[T2], 
For[k=1, k≤n,
If[MemberQ[T2[[j]],Mod[k,n]],H[[j,k]]=1];
k++];
j++];

MF[[1]]=H[[1]];
MT[[1]]=T2[[1]];

cont=1;
v=1;
For[j=2, j≤Length[T2], cont=1;
For[c=1,c≤v,
For[k=1, k≤n,
If[(Not[Chop[Fourier[MF[[c]]]][[k]]==Chop[Fourier[H[[j]]]][[k]]])∧(Chop[Fourier[MF[[c]]]][[k]]==0∨Chop[Fourier[H[[j]]]][[k]]==0), 
k=n+1;cont=cont+1];
k++];
If[cont≤c, c=v+1];
c++];
If[cont==v+1, MF[[v+1]]=H[[j]];MT[[v+1]]=T2[[j]]; v=v+1];
j++]; 


MT=Drop[MT⋃{},1];
MF=Drop[MF⋃{},1];
Length[MF]



MT

(*{{3,8,16},{2,4,9}};*)

(* Next, we see what those Fourier-zero sets are. *)

n=144;
J=MF;
F1=Array[1&,{Length[J],n}];

For[i=1,i<=Length[J],
For[j=1,j≤n,
If[Chop[Fourier[J[[i]]]][[j]]==0, F1[[i,j]]=0];
j++];
i++];

F1;


n=144;
FF1=Array[0&,{Length[F1],n}];

For[i=1,i≤Length[F1],
For[j=1,j≤n,
If[F1[[i,j]]==0, FF1[[i,j]]=j];
j++];
i++];

For[j=1,j≤Length[FF1],
FF1[[j]]=Drop[FF1[[j]]⋃{},1];
j++];
FF1;


FF=FF1-1;

For[m=1,m≤Length[FF1],
For[j=1,j≤Length[FF1[[m]]],
FF[[m,j]]=GCD[FF[[m,j]],n];
j++];
m++];
FF;
For[j=1,j≤Length[FF],
FF[[j]]=FF[[j]]⋃{};
j++];
FF

FT=n/FF
CFT=Array[0&,Length[FT]];
For[j=1,j≤Length[FT],
CFT[[j]]=Complement[Divisors[n],FT[[j]]];
j++];
CFT

T2={{0,34,40,46,48,58,88,96,106,118,130,136},{0,34,42,48,60,72,82,90,120,130,132,138},{0,38,40,48,50,62,88,96,110,122,134,136},{0,38,46,48,60,72,86,94,120,132,134,142},{0,40,42,48,54,66,88,96,114,126,136,138},{0,40,46,48,58,70,88,96,118,130,136,142}};

Clear[H];

Length[T2]

n=144;
H=Array[0&,{Length[T2],n}];
MF=Array[0&,100];
MT=Array[0&,100];

For[j=1, j≤Length[T2], 
For[k=1, k≤n,
If[MemberQ[T2[[j]],Mod[k,n]],H[[j,k]]=1];
k++];
j++];

MF[[1]]=H[[1]];
MT[[1]]=T2[[1]];

cont=1;
v=1;
For[j=2, j≤Length[T2], cont=1;
For[c=1,c≤v,
For[k=1, k≤n,
If[(Not[Chop[Fourier[MF[[c]]]][[k]]==Chop[Fourier[H[[j]]]][[k]]])∧(Chop[Fourier[MF[[c]]]][[k]]==0∨Chop[Fourier[H[[j]]]][[k]]==0), 
k=n+1;cont=cont+1];
k++];
If[cont≤c, c=v+1];
c++];
If[cont==v+1, MF[[v+1]]=H[[j]];MT[[v+1]]=T2[[j]]; v=v+1];
j++]; 


MT=Drop[MT⋃{},1];
MF=Drop[MF⋃{},1];
Length[MF]



MT

(*{{3,8,16},{2,4,9}};*)

(* Next, we see what those Fourier-zero sets are. *)

n=144;
J=MF;
F1=Array[1&,{Length[J],n}];

For[i=1,i<=Length[J],
For[j=1,j≤n,
If[Chop[Fourier[J[[i]]]][[j]]==0, F1[[i,j]]=0];
j++];
i++];

F1;


n=144;
FF1=Array[0&,{Length[F1],n}];

For[i=1,i≤Length[F1],
For[j=1,j≤n,
If[F1[[i,j]]==0, FF1[[i,j]]=j];
j++];
i++];

For[j=1,j≤Length[FF1],
FF1[[j]]=Drop[FF1[[j]]⋃{},1];
j++];
FF1;


FF=FF1-1;

For[m=1,m≤Length[FF1],
For[j=1,j≤Length[FF1[[m]]],
FF[[m,j]]=GCD[FF[[m,j]],n];
j++];
m++];
FF;
For[j=1,j≤Length[FF],
FF[[j]]=FF[[j]]⋃{};
j++];
FF

FT=n/FF
CFT=Array[0&,Length[FT]];
For[j=1,j≤Length[FT],
CFT[[j]]=Complement[Divisors[n],FT[[j]]];
j++];
CFT