Properties

Modulus 87
Structure \(C_{28}\times C_{2}\)
Order 56

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Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(87)
 
pari: g = idealstar(,87,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 56
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{28}\times C_{2}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{87}(31,\cdot)$, $\chi_{87}(59,\cdot)$

First 32 of 56 characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

orbit label order primitive -1 1 2 4 5 7 8 10 11 13 14 16
\(\chi_{87}(1,\cdot)\) 87.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{87}(2,\cdot)\) 87.k 28 Yes \(1\) \(1\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{87}(4,\cdot)\) 87.i 14 No \(1\) \(1\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{87}(5,\cdot)\) 87.h 14 Yes \(-1\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{87}(7,\cdot)\) 87.g 7 No \(1\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{87}(8,\cdot)\) 87.k 28 Yes \(1\) \(1\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{87}(10,\cdot)\) 87.l 28 No \(-1\) \(1\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{87}(11,\cdot)\) 87.k 28 Yes \(1\) \(1\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{4}{7}\right)\)
\(\chi_{87}(13,\cdot)\) 87.i 14 No \(1\) \(1\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{4}{7}\right)\)
\(\chi_{87}(14,\cdot)\) 87.k 28 Yes \(1\) \(1\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{87}(16,\cdot)\) 87.g 7 No \(1\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{4}{7}\right)\)
\(\chi_{87}(17,\cdot)\) 87.f 4 Yes \(1\) \(1\) \(i\) \(-1\) \(1\) \(1\) \(-i\) \(i\) \(i\) \(-1\) \(i\) \(1\)
\(\chi_{87}(19,\cdot)\) 87.l 28 No \(-1\) \(1\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{87}(20,\cdot)\) 87.j 14 Yes \(-1\) \(1\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{87}(22,\cdot)\) 87.i 14 No \(1\) \(1\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{87}(23,\cdot)\) 87.j 14 Yes \(-1\) \(1\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{87}(25,\cdot)\) 87.g 7 No \(1\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{87}(26,\cdot)\) 87.k 28 Yes \(1\) \(1\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{87}(28,\cdot)\) 87.c 2 No \(1\) \(1\) \(-1\) \(1\) \(1\) \(1\) \(-1\) \(-1\) \(-1\) \(1\) \(-1\) \(1\)
\(\chi_{87}(31,\cdot)\) 87.l 28 No \(-1\) \(1\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{87}(32,\cdot)\) 87.k 28 Yes \(1\) \(1\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{87}(34,\cdot)\) 87.i 14 No \(1\) \(1\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{87}(35,\cdot)\) 87.h 14 Yes \(-1\) \(1\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{87}(37,\cdot)\) 87.l 28 No \(-1\) \(1\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{87}(38,\cdot)\) 87.h 14 Yes \(-1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{87}(40,\cdot)\) 87.l 28 No \(-1\) \(1\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{4}{7}\right)\)
\(\chi_{87}(41,\cdot)\) 87.f 4 Yes \(1\) \(1\) \(-i\) \(-1\) \(1\) \(1\) \(i\) \(-i\) \(-i\) \(-1\) \(-i\) \(1\)
\(\chi_{87}(43,\cdot)\) 87.l 28 No \(-1\) \(1\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{87}(44,\cdot)\) 87.k 28 Yes \(1\) \(1\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{87}(46,\cdot)\) 87.e 4 No \(-1\) \(1\) \(-i\) \(-1\) \(-1\) \(1\) \(i\) \(i\) \(-i\) \(-1\) \(-i\) \(1\)
\(\chi_{87}(47,\cdot)\) 87.k 28 Yes \(1\) \(1\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{4}{7}\right)\)
\(\chi_{87}(49,\cdot)\) 87.g 7 No \(1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{7}\right)\)