Properties

Modulus 97
Structure \(C_{96}\)
Order 96

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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(97)
 
pari: g = idealstar(,97,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 96
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{96}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{97}(5,\cdot)$

First 32 of 96 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 3 4 5 6 7 8 9 10 11
\(\chi_{97}(1,\cdot)\) 97.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{97}(2,\cdot)\) 97.k 48 Yes \(1\) \(1\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{11}{24}\right)\)
\(\chi_{97}(3,\cdot)\) 97.k 48 Yes \(1\) \(1\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{17}{24}\right)\)
\(\chi_{97}(4,\cdot)\) 97.i 24 Yes \(1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{23}{24}\right)\) \(i\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{11}{12}\right)\)
\(\chi_{97}(5,\cdot)\) 97.l 96 Yes \(-1\) \(1\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{1}{96}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{31}{96}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{35}{96}\right)\) \(e\left(\frac{43}{48}\right)\)
\(\chi_{97}(6,\cdot)\) 97.g 12 Yes \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{97}(7,\cdot)\) 97.l 96 Yes \(-1\) \(1\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{31}{96}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{96}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{29}{96}\right)\) \(e\left(\frac{37}{48}\right)\)
\(\chi_{97}(8,\cdot)\) 97.h 16 Yes \(1\) \(1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(i\) \(e\left(\frac{1}{16}\right)\) \(-1\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(-i\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{3}{8}\right)\)
\(\chi_{97}(9,\cdot)\) 97.i 24 Yes \(1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{24}\right)\) \(-i\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{97}(10,\cdot)\) 97.l 96 Yes \(-1\) \(1\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{35}{96}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{29}{96}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{73}{96}\right)\) \(e\left(\frac{17}{48}\right)\)
\(\chi_{97}(11,\cdot)\) 97.k 48 Yes \(1\) \(1\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{1}{24}\right)\)
\(\chi_{97}(12,\cdot)\) 97.h 16 Yes \(1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(-i\) \(e\left(\frac{7}{16}\right)\) \(-1\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(i\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{5}{8}\right)\)
\(\chi_{97}(13,\cdot)\) 97.l 96 Yes \(-1\) \(1\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{25}{96}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{7}{96}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{11}{96}\right)\) \(e\left(\frac{19}{48}\right)\)
\(\chi_{97}(14,\cdot)\) 97.l 96 Yes \(-1\) \(1\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{65}{96}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{95}{96}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{67}{96}\right)\) \(e\left(\frac{11}{48}\right)\)
\(\chi_{97}(15,\cdot)\) 97.l 96 Yes \(-1\) \(1\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{71}{96}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{89}{96}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{85}{96}\right)\) \(e\left(\frac{29}{48}\right)\)
\(\chi_{97}(16,\cdot)\) 97.g 12 Yes \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{97}(17,\cdot)\) 97.l 96 Yes \(-1\) \(1\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{89}{96}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{71}{96}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{43}{96}\right)\) \(e\left(\frac{35}{48}\right)\)
\(\chi_{97}(18,\cdot)\) 97.h 16 Yes \(1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(i\) \(e\left(\frac{13}{16}\right)\) \(-1\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(-i\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{7}{8}\right)\)
\(\chi_{97}(19,\cdot)\) 97.j 32 Yes \(-1\) \(1\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{27}{32}\right)\) \(-i\) \(e\left(\frac{5}{32}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{17}{32}\right)\) \(e\left(\frac{9}{16}\right)\)
\(\chi_{97}(20,\cdot)\) 97.j 32 Yes \(-1\) \(1\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{23}{32}\right)\) \(-i\) \(e\left(\frac{9}{32}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{5}{32}\right)\) \(e\left(\frac{13}{16}\right)\)
\(\chi_{97}(21,\cdot)\) 97.l 96 Yes \(-1\) \(1\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{5}{96}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{59}{96}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{79}{96}\right)\) \(e\left(\frac{23}{48}\right)\)
\(\chi_{97}(22,\cdot)\) 97.d 4 Yes \(1\) \(1\) \(-1\) \(-1\) \(1\) \(i\) \(1\) \(-i\) \(-1\) \(1\) \(-i\) \(-1\)
\(\chi_{97}(23,\cdot)\) 97.l 96 Yes \(-1\) \(1\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{77}{96}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{83}{96}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{7}{96}\right)\) \(e\left(\frac{47}{48}\right)\)
\(\chi_{97}(24,\cdot)\) 97.i 24 Yes \(1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{13}{24}\right)\) \(-i\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{97}(25,\cdot)\) 97.k 48 Yes \(1\) \(1\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{19}{24}\right)\)
\(\chi_{97}(26,\cdot)\) 97.l 96 Yes \(-1\) \(1\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{59}{96}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{96}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{49}{96}\right)\) \(e\left(\frac{41}{48}\right)\)
\(\chi_{97}(27,\cdot)\) 97.h 16 Yes \(1\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(-i\) \(e\left(\frac{3}{16}\right)\) \(-1\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(i\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{1}{8}\right)\)
\(\chi_{97}(28,\cdot)\) 97.j 32 Yes \(-1\) \(1\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{1}{32}\right)\) \(i\) \(e\left(\frac{31}{32}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{3}{32}\right)\) \(e\left(\frac{11}{16}\right)\)
\(\chi_{97}(29,\cdot)\) 97.l 96 Yes \(-1\) \(1\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{13}{96}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{19}{96}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{71}{96}\right)\) \(e\left(\frac{31}{48}\right)\)
\(\chi_{97}(30,\cdot)\) 97.j 32 Yes \(-1\) \(1\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{3}{32}\right)\) \(-i\) \(e\left(\frac{29}{32}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{9}{32}\right)\) \(e\left(\frac{1}{16}\right)\)
\(\chi_{97}(31,\cdot)\) 97.k 48 Yes \(1\) \(1\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{5}{24}\right)\)
\(\chi_{97}(32,\cdot)\) 97.k 48 Yes \(1\) \(1\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{7}{24}\right)\)