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However it works as well with negative lengths; in this case the spokes \ representing vectors are sticking backward from the direction.\ \>", "Text"], Cell[BoxData[ \(\(coeffList = { .2, .8, \(- .2\), .6, \(- .9\), 1, \(- .6\), .4, \(- .6\), .5, \(- .9\), 1, .3};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(% // TableForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {"0.2`"}, {"0.8`"}, {\(-0.2`\)}, {"0.6`"}, {\(-0.9`\)}, {"1"}, {\(-0.6`\)}, {"0.4`"}, {\(-0.6`\)}, {"0.5`"}, {\(-0.9`\)}, {"1"}, {"0.3`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {0.20000000000000001, 0.80000000000000004, -0.20000000000000001, 0.59999999999999998, -0.90000000000000002, 1, -0.59999999999999998, 0.40000000000000002, -0.59999999999999998, 0.5, -0.90000000000000002, 1, 0.29999999999999999}]]], "Output"] }, Closed]], Cell["For later reference, here is their sum:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plus @@ coeffList\)], "Input"], Cell[BoxData[ \(1.6000000000000003`\)], "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Start by using a fixed prime number ", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(n = \ Length[coeffList]\)], "Input"], Cell[BoxData[ \(13\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell["For esthetical reasons we use a prime number of directions", \ "Subsubsection"], Cell["This is necessary to have the visual representations below.", "Text"], Cell["\<\ Remark: we will explain later what is different if n is no \ longer a prime. 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called the ", StyleBox["Discrete Fourier transform", FontVariations->{"Underline"->True}], " of coeffList." }], "Text", FontWeight->"Bold"], Cell[TextData[{ "The command for obtaining this DFT is called ", StyleBox["Fourier", FontWeight->"Bold"], " in Mathematica ." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Fourier[coeffList] // TableForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(\(\(0.44376015698018334`\)\(\[InvisibleSpace]\)\) + 0.`\ \[ImaginaryI]\)}, {\(\(\(0.45178068930694143`\)\(\[InvisibleSpace]\)\) + 0.06836738362651347`\ \[ImaginaryI]\)}, {\(\(\(0.27818148921177477`\)\(\[InvisibleSpace]\)\) - 0.15165319393494747`\ \[ImaginaryI]\)}, {\(\(-0.08737526591999054`\) + 0.03076235378510685`\ \[ImaginaryI]\)}, {\(\(-0.4450678988587973`\) - 0.10637570741199508`\ \[ImaginaryI]\)}, {\(\(\(0.03220817094661013`\)\(\[InvisibleSpace]\)\) + 0.39210547317587374`\ \[ImaginaryI]\)}, {\(\(-0.09105213563023118`\) + 1.4735394103969923`\ \[ImaginaryI]\)}, {\(\(-0.09105213563023118`\) - 1.4735394103969923`\ \[ImaginaryI]\)}, {\(\(\(0.03220817094661013`\)\(\[InvisibleSpace]\)\) - 0.39210547317587374`\ \[ImaginaryI]\)}, {\(\(-0.4450678988587973`\) + 0.10637570741199508`\ \[ImaginaryI]\)}, {\(\(-0.08737526591999054`\) - 0.03076235378510685`\ \[ImaginaryI]\)}, {\(\(\(0.27818148921177477`\)\(\[InvisibleSpace]\)\) + 0.15165319393494747`\ \[ImaginaryI]\)}, {\(\(\(0.45178068930694143`\)\(\[InvisibleSpace]\)\) - 0.06836738362651347`\ \[ImaginaryI]\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ { Complex[ 0.44376015698018334, 0.0], Complex[ 0.45178068930694143, 0.06836738362651347], Complex[ 0.27818148921177477, -0.15165319393494747], Complex[ -0.087375265919990536, 0.03076235378510685], Complex[ -0.44506789885879727, -0.10637570741199508], Complex[ 0.032208170946610132, 0.39210547317587374], Complex[ -0.091052135630231179, 1.4735394103969923], Complex[ -0.091052135630231179, -1.4735394103969923], Complex[ 0.032208170946610132, -0.39210547317587374], Complex[ -0.44506789885879727, 0.10637570741199508], Complex[ -0.087375265919990536, -0.03076235378510685], Complex[ 0.27818148921177477, 0.15165319393494747], Complex[ 0.45178068930694143, -0.06836738362651347]}]]], "Output"] }, Closed]], Cell["Here are the sums we have obtained:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Chop[\@n\ %\ ] // TableForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {"1.6`"}, {\(\(\(1.628918440560643`\)\(\[InvisibleSpace]\)\) + 0.2465021072347115`\ \[ImaginaryI]\)}, {\(\(\(1.0029976232379865`\)\(\[InvisibleSpace]\)\) - 0.5467933668203375`\ \[ImaginaryI]\)}, {\(\(-0.3150360014818271`\) + 0.11091524392616647`\ \[ImaginaryI]\)}, {\(\(-1.6047151303984142`\) - 0.38354306753770295`\ \[ImaginaryI]\)}, {\(\(\(0.11612821183691235`\)\(\[InvisibleSpace]\)\) + 1.4137563889256826`\ \[ImaginaryI]\)}, {\(\(-0.32829314375530017`\) + 5.31292190060333`\ \[ImaginaryI]\)}, {\(\(-0.32829314375530017`\) - 5.31292190060333`\ \[ImaginaryI]\)}, {\(\(\(0.11612821183691235`\)\(\[InvisibleSpace]\)\) - 1.4137563889256826`\ \[ImaginaryI]\)}, {\(\(-1.6047151303984142`\) + 0.38354306753770295`\ \[ImaginaryI]\)}, {\(\(-0.3150360014818271`\) - 0.11091524392616647`\ \[ImaginaryI]\)}, {\(\(\(1.0029976232379865`\)\(\[InvisibleSpace]\)\) + 0.5467933668203375`\ \[ImaginaryI]\)}, {\(\(\(1.628918440560643`\)\(\[InvisibleSpace]\)\) - 0.2465021072347115`\ \[ImaginaryI]\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {1.6000000000000001, Complex[ 1.6289184405606429, 0.2465021072347115], Complex[ 1.0029976232379865, -0.54679336682033752], Complex[ -0.31503600148182709, 0.11091524392616647], Complex[ -1.6047151303984142, -0.38354306753770295], Complex[ 0.11612821183691235, 1.4137563889256826], Complex[ -0.32829314375530017, 5.3129219006033299], Complex[ -0.32829314375530017, -5.3129219006033299], Complex[ 0.11612821183691235, -1.4137563889256826], Complex[ -1.6047151303984142, 0.38354306753770295], Complex[ -0.31503600148182709, -0.11091524392616647], Complex[ 1.0029976232379865, 0.54679336682033752], Complex[ 1.6289184405606429, -0.2465021072347115]}]]], "Output"] }, Closed]], Cell["\<\ It is possible to check coordinates by hand in some of the graphs \ above.\ \>", "Text"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Recovering the original coefficient List from these sums.", "Section"], Cell[CellGroupData[{ Cell["Introduction", "Subsection"], Cell["\<\ We may ask now whether given all these n sums we can recover \ the original entries in the original list we called coeffList. It boils down \ to solve an n x n system of linear equations with complex coefficients. \ Indeed, the coefficients in the system of equations are the n directions \ in the plane. These are repeated in each equation in a different order \ according to the way of counting in the harmonics. Solving such a system \ looks like a huge task, and at this point the reader may not know yet how to \ perform this. Only to know that a unique solution exists, one has to compute \ a n x n determinant of all directions and be sure this determinant is not \ zero. Fortunately, we can do this in a different way. The only thing we have to know is the sum of the geometric progression at the \ start of this text.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Reversing directions", "Subsection"], Cell["\<\ Recall that for the k harmonic we used directions in the \ following order\ \>", "Text"], Cell[BoxData[ \(mixDirections[n_, k_] := \[ExponentialE]\^\(\[ImaginaryI]\ k\ Range[n]\ 2\ \ \(\(\[Pi]/n\)\(\ \)\)\)\)], "Input"], Cell["obtaining the k th sum. Let us write all these sums as", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Array[s, {n}]\)], "Input"], Cell[BoxData[ \({s[1], s[2], s[3], s[4], s[5], s[6], s[7], s[8], s[9], s[10], s[11], s[12], s[13]}\)], "Output"] }, Closed]], Cell["\<\ and use them as a new list of \"lengths\" replacing coeffList. These we are going to multiply by directions measured clockwise around the \ origin. Since these sums are no longer real numbers, the vectors obtained do \ not point along direction (or in the opposite direction) but may point \ anywhere. Drawing a graph gives no information, but we can still add these vectors in \ the plane and obtain new sums. For the j th harmonic, we use \ \>", "Text"], Cell[BoxData[ \(mixDirections[n, \(-\ j\)]\)], "Input", Evaluatable->False], Cell["e.g. ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(j = 4; mixDirections[n, \(-\ j\)]\)], "Input"], Cell[BoxData[ \({\[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\), \ \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\), \[ExponentialE]\^\(\(2\ \ \[ImaginaryI]\ \[Pi]\)\/13\), \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \ \[Pi]\)\/13\)\), \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\), \ \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\), \ \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\), \ \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\), \[ExponentialE]\^\ \(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\), \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \ \[Pi]\)\/13\)\), \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \ \[Pi]\)\/13\)\), \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\), 1}\)], "Output"] }, Closed]], Cell["The sum corresponding to this 4 th harmonic is ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Array[s, {n}] . mixDirections[n, \(-\ j\)]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ s[ 1] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ s[ 2] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ s[ 3] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ s[ 4] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ s[ 5] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ s[ 6] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ s[ 7] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ s[ 8] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ s[ 9] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ s[ 10] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ s[ 11] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ s[12] + s[13]\)], "Output"] }, Closed]], Cell["\<\ The result will be n times the j th entry in \ coeffList.\ \>", "Text", FontWeight->"Bold"] }, Closed]], Cell["Experimenting", "Subsection"], Cell[CellGroupData[{ Cell["Proving this result", "Subsection"], Cell["Start with any coeffList", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(coeffList = Array[c, {n}]\)], "Input"], Cell[BoxData[ \({c[1], c[2], c[3], c[4], c[5], c[6], c[7], c[8], c[9], c[10], c[11], c[12], c[13]}\)], "Output"] }, Closed]], Cell["Replace each s[k] by its value and write all new sums", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[coeffList . mixDirections[n, k], {k, n}]\)], "Input"], Cell[BoxData[ \({\[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 1] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 2] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 3] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 4] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 5] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 6] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[7] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 8] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 9] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 10] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[11] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[12] + c[13], \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 1] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 2] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 3] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[4] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 5] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 6] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 7] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 8] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 9] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[10] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 11] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[12] + c[ 13], \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 1] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 2] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 3] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 4] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 5] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 6] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[7] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 8] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 9] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 10] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[11] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[12] + c[13], \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 1] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[2] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 3] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 4] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[5] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 6] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 7] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 8] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 9] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 10] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 11] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[12] + c[ 13], \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 1] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 2] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 3] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[4] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 5] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 6] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 7] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 8] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 9] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 10] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 11] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[12] + c[ 13], \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 1] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 2] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 3] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 4] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 5] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 6] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 7] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 8] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 9] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[10] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 11] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[12] + c[ 13], \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 1] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 2] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[3] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 4] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 5] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 6] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 7] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 8] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 9] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 10] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[11] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[12] + c[13], \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 1] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 2] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 3] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 4] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 5] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 6] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 7] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 8] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[9] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 10] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[11] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[12] + c[13], \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 1] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 2] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 3] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 4] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 5] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 6] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 7] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[8] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 9] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 10] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[11] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[12] + c[13], \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 1] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[2] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 3] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 4] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 5] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[6] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 7] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 8] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 9] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 10] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 11] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 12] + c[13], \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \ \[Pi]\)\/13\)\)\ c[ 1] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 2] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[3] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 4] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 5] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 6] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 7] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 8] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[9] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 10] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 11] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 12] + c[13], \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \ \[Pi]\)\/13\)\)\ c[ 1] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 2] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 3] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 4] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ \ c[5] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 6] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 7] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 8] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 9] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 10] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 11] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 12] + c[13], c[1] + c[2] + c[3] + c[4] + c[5] + c[6] + c[7] + c[8] + c[9] + c[10] + c[11] + c[12] + c[13]}\)], "Output"] }, Closed]], Cell["\<\ (Recall that we set j = 4 , but the following can be repeated \ for any j )\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[% . mixDirections[n, \(-\ j\)]]\)], "Input"], Cell[BoxData[ \(c[1] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 1] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 1] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 1] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 1] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 1] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 1] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 1] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 1] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 1] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 1] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 1] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[1] + c[2] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 2] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 2] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 2] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 2] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 2] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 2] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 2] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 2] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 2] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 2] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 2] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[2] + c[3] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 3] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 3] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 3] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 3] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 3] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 3] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 3] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 3] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 3] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 3] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 3] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[3] + 13\ c[4] + c[5] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 5] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 5] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 5] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 5] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 5] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 5] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 5] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 5] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 5] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 5] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 5] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[5] + c[6] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 6] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 6] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 6] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 6] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 6] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 6] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 6] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 6] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 6] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 6] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 6] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[6] + c[7] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 7] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 7] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 7] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 7] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 7] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 7] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 7] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 7] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 7] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 7] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 7] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[7] + c[8] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 8] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 8] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 8] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 8] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 8] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 8] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 8] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 8] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 8] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 8] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 8] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[8] + c[9] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 9] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 9] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 9] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 9] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 9] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 9] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 9] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 9] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 9] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 9] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 9] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[9] + c[10] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 10] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 10] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 10] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 10] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 10] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 10] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 10] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 10] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 10] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 10] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 10] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 10] + c[11] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\ \)\)\ c[11] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 11] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 11] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 11] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 11] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 11] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 11] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 11] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 11] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 11] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 11] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 11] + c[12] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\ \)\)\ c[12] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 12] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 12] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 12] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 12] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 12] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 12] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 12] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 12] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 12] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 12] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 12] + c[13] + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\ \)\)\ c[13] + \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 13] + \[ExponentialE]\^\(-\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 13] + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 13] + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 13] + \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 13] + \[ExponentialE]\^\(-\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 13] + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 13] + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 13] + \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 13] + \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\)\ c[ 13] + \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\ c[ 13]\)], "Output"] }, Closed]], Cell["\<\ We group according to the numbering of the c and make a list of \ coefficients \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[Coefficient[%, c[m]], {m, n}]\)], "Input"], Cell[BoxData[ \({1 + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\) + \ \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\), 1 + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\) + \ \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\), 1 + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\) + \ \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\), 13, 1 + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\) + \ \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\), 1 + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\) + \ \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\), 1 + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\) + \ \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\), 1 + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ 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\[Pi]\)\/13\) + \ \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\), 1 + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\) + \ \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\), 1 + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \[ExponentialE]\^\(\(8\ \[ImaginaryI]\ \ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(10\ \[ImaginaryI]\ \[Pi]\)\/13\) + \ \[ExponentialE]\^\(-\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(12\ \[ImaginaryI]\ \[Pi]\)\/13\), 1 + \[ExponentialE]\^\(-\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(2\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(4\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \[ExponentialE]\^\(\(4\ \[ImaginaryI]\ \ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + \ \[ExponentialE]\^\(\(6\ \[ImaginaryI]\ \[Pi]\)\/13\) + \[ExponentialE]\^\(-\(\ \(8\ \[ImaginaryI]\ \[Pi]\)\/13\)\) + 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