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In[1]:=  			SALEM 12:  Twisting at 7

         Ideal Theory

                 2    3    6    9    10    11    12         3
GB for {1 - x + x  - x  - x  - x  + x   - x   + x  , (1 + x) }: 
 
                               2
>   {49, 7 + 7 x, -20 + 2 x + x }

              7               2                               2
GB for {-1 + u , 2 + 3 u + 2 u }: {49, -7 + 7 u, -20 - 2 u + u }

             2   3                  2
GB for {7 + t , t }: {49, 7 t, 7 + t }

                                      2                              2
GB for {49, 7 (3 + 3 x), 7 + (3 + 3 x) }: {49, 7 + 7 x, -20 + 2 x + x }
                              2
 = {49, 7 + 7 x, -20 + 2 x + x }

                                   2                              2
GB for {49, -7 - 7 x, -20 + 2 x + x }: {49, 7 + 7 x, -20 + 2 x + x }
                              2
 = {49, 7 + 7 x, -20 + 2 x + x }
                                    2      3    4    5
Fixing signature; twist is 3 x + 2 x  - 4 x  - x  + x

 			Salem Factor
Glue group {7, 49}, Signature {11, 1}, lambda 1.24073

Positivity verified.  Crossing: 4 Cyclic: Infinity Fixed: Infinity
                    2    3    6    9    10    11    12
Char poly  1 - x + x  - x  - x  - x  + x   - x   + x
                          2
Localization at 7: (1 + x)
                                                             2
At prime 7 : Glue group {7, 49}, Period 14, Char poly (1 + x)
                 q = 1/49 * 14   21   f = 6    3

                            21   8        7    48

 			Cyclotomic Factor 

Glue group {7, 49}, Signature {5, 2}, lambda = 1.
                              2    3    4    5    6
Char poly  (-1 + x) (1 + x + x  + x  + x  + x  + x )  Period 1
                          2
Localization at 7: (6 + x)
                                                            2
At prime 7 : Glue group {7, 49}, Period 7, Char poly (6 + x)
                 q = 1/49 * 14   14   f = 1    3

                            14   29       7    43
For compatible gluing these should be opposite:  1 != 1
Glue group {7, 49}, Signature {5, 2}, lambda = 1.
                              2    3    4    5    6
Char poly  (-1 + x) (1 + x + x  + x  + x  + x  + x )  Period 1
                          2
Localization at 7: (6 + x)
                                                            2
At prime 7 : Glue group {7, 49}, Period 7, Char poly (6 + x)
                 q = 1/49 * 14   14   f = 1    3

                            14   29       7    43
-----------------------------EXTRAS--------------------
Number of extras is 237
We will sample a few
                       2       3
Extra: -8 + 14 x + 14 x  - 11 x  Sig {18, 1} Glue {4, 27} Positive? False
                       2      3
Extra: -6 + 10 x + 12 x  - 9 x  Sig {18, 1} Glue {2, 47} Positive? False
Some extra worked?? False
Glue group {7, 49}, Signature {5, 2}, lambda = 1.
                              2    3    4    5    6
Char poly  (-1 + x) (1 + x + x  + x  + x  + x  + x )  Period 1
                          2
Localization at 7: (6 + x)
                                                            2
At prime 7 : Glue group {7, 49}, Period 7, Char poly (6 + x)
                 q = 1/49 * 14   14   f = 1    3

                            14   29       7    43
              2
EXTRA is 2 - x
Glue group {2}, Signature {17, 2}, lambda = 1.24073
                             2    3    4    5    6
Char poly  (1 + x) (1 - x + x  - x  + x  - x  + x ) 
 
               2    3    6    9    10    11    12
>    (1 - x + x  - x  - x  - x  + x   - x   + x  )
Localization at 2: 1 + x
At prime 2 : Glue group {2}, Period 1, Char poly 1 + x
                 q = 1/2 * 1   f = 1

 			Realized (but wrong signature)
Glue group {}, Signature {20, 2}, lambda = 1.24073
                   3                   2    3    4    5    6
Char poly  (-1 + x)  (1 + x) (1 - x + x  - x  + x  - x  + x ) 
 
               2    3    6    9    10    11    12
>    (1 - x + x  - x  - x  - x  + x   - x   + x  )
Unimodular

In[2]:= 