Properties

Modulus 163
Structure \(C_{162}\)
Order 162

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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(163)
 
pari: g = idealstar(,163,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 162
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{162}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{163}(2,\cdot)$

First 32 of 162 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
\(\chi_{163}(1,\cdot)\) 163.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{163}(2,\cdot)\) 163.j 162 Yes \(-1\) \(1\) \(e\left(\frac{1}{162}\right)\) \(e\left(\frac{101}{162}\right)\) \(e\left(\frac{1}{81}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{73}{162}\right)\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{20}{81}\right)\) \(e\left(\frac{8}{81}\right)\) \(e\left(\frac{47}{162}\right)\)
\(\chi_{163}(3,\cdot)\) 163.j 162 Yes \(-1\) \(1\) \(e\left(\frac{101}{162}\right)\) \(e\left(\frac{157}{162}\right)\) \(e\left(\frac{20}{81}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{83}{162}\right)\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{76}{81}\right)\) \(e\left(\frac{79}{81}\right)\) \(e\left(\frac{49}{162}\right)\)
\(\chi_{163}(4,\cdot)\) 163.i 81 Yes \(1\) \(1\) \(e\left(\frac{1}{81}\right)\) \(e\left(\frac{20}{81}\right)\) \(e\left(\frac{2}{81}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{73}{81}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{40}{81}\right)\) \(e\left(\frac{16}{81}\right)\) \(e\left(\frac{47}{81}\right)\)
\(\chi_{163}(5,\cdot)\) 163.h 54 Yes \(-1\) \(1\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{19}{54}\right)\)
\(\chi_{163}(6,\cdot)\) 163.g 27 Yes \(1\) \(1\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{16}{27}\right)\)
\(\chi_{163}(7,\cdot)\) 163.j 162 Yes \(-1\) \(1\) \(e\left(\frac{73}{162}\right)\) \(e\left(\frac{83}{162}\right)\) \(e\left(\frac{73}{81}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{145}{162}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{2}{81}\right)\) \(e\left(\frac{17}{81}\right)\) \(e\left(\frac{29}{162}\right)\)
\(\chi_{163}(8,\cdot)\) 163.h 54 Yes \(-1\) \(1\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{47}{54}\right)\)
\(\chi_{163}(9,\cdot)\) 163.i 81 Yes \(1\) \(1\) \(e\left(\frac{20}{81}\right)\) \(e\left(\frac{76}{81}\right)\) \(e\left(\frac{40}{81}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{2}{81}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{71}{81}\right)\) \(e\left(\frac{77}{81}\right)\) \(e\left(\frac{49}{81}\right)\)
\(\chi_{163}(10,\cdot)\) 163.i 81 Yes \(1\) \(1\) \(e\left(\frac{8}{81}\right)\) \(e\left(\frac{79}{81}\right)\) \(e\left(\frac{16}{81}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{17}{81}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{77}{81}\right)\) \(e\left(\frac{47}{81}\right)\) \(e\left(\frac{52}{81}\right)\)
\(\chi_{163}(11,\cdot)\) 163.j 162 Yes \(-1\) \(1\) \(e\left(\frac{47}{162}\right)\) \(e\left(\frac{49}{162}\right)\) \(e\left(\frac{47}{81}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{29}{162}\right)\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{49}{81}\right)\) \(e\left(\frac{52}{81}\right)\) \(e\left(\frac{103}{162}\right)\)
\(\chi_{163}(12,\cdot)\) 163.j 162 Yes \(-1\) \(1\) \(e\left(\frac{103}{162}\right)\) \(e\left(\frac{35}{162}\right)\) \(e\left(\frac{22}{81}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{67}{162}\right)\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{35}{81}\right)\) \(e\left(\frac{14}{81}\right)\) \(e\left(\frac{143}{162}\right)\)
\(\chi_{163}(13,\cdot)\) 163.h 54 Yes \(-1\) \(1\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{43}{54}\right)\)
\(\chi_{163}(14,\cdot)\) 163.i 81 Yes \(1\) \(1\) \(e\left(\frac{37}{81}\right)\) \(e\left(\frac{11}{81}\right)\) \(e\left(\frac{74}{81}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{28}{81}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{22}{81}\right)\) \(e\left(\frac{25}{81}\right)\) \(e\left(\frac{38}{81}\right)\)
\(\chi_{163}(15,\cdot)\) 163.i 81 Yes \(1\) \(1\) \(e\left(\frac{58}{81}\right)\) \(e\left(\frac{26}{81}\right)\) \(e\left(\frac{35}{81}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{22}{81}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{52}{81}\right)\) \(e\left(\frac{37}{81}\right)\) \(e\left(\frac{53}{81}\right)\)
\(\chi_{163}(16,\cdot)\) 163.i 81 Yes \(1\) \(1\) \(e\left(\frac{2}{81}\right)\) \(e\left(\frac{40}{81}\right)\) \(e\left(\frac{4}{81}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{65}{81}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{80}{81}\right)\) \(e\left(\frac{32}{81}\right)\) \(e\left(\frac{13}{81}\right)\)
\(\chi_{163}(17,\cdot)\) 163.h 54 Yes \(-1\) \(1\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{29}{54}\right)\)
\(\chi_{163}(18,\cdot)\) 163.j 162 Yes \(-1\) \(1\) \(e\left(\frac{41}{162}\right)\) \(e\left(\frac{91}{162}\right)\) \(e\left(\frac{41}{81}\right)\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{77}{162}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{10}{81}\right)\) \(e\left(\frac{4}{81}\right)\) \(e\left(\frac{145}{162}\right)\)
\(\chi_{163}(19,\cdot)\) 163.j 162 Yes \(-1\) \(1\) \(e\left(\frac{125}{162}\right)\) \(e\left(\frac{151}{162}\right)\) \(e\left(\frac{44}{81}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{53}{162}\right)\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{70}{81}\right)\) \(e\left(\frac{28}{81}\right)\) \(e\left(\frac{43}{162}\right)\)
\(\chi_{163}(20,\cdot)\) 163.j 162 Yes \(-1\) \(1\) \(e\left(\frac{17}{162}\right)\) \(e\left(\frac{97}{162}\right)\) \(e\left(\frac{17}{81}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{107}{162}\right)\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{16}{81}\right)\) \(e\left(\frac{55}{81}\right)\) \(e\left(\frac{151}{162}\right)\)
\(\chi_{163}(21,\cdot)\) 163.g 27 Yes \(1\) \(1\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{13}{27}\right)\)
\(\chi_{163}(22,\cdot)\) 163.g 27 Yes \(1\) \(1\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{25}{27}\right)\)
\(\chi_{163}(23,\cdot)\) 163.f 18 Yes \(-1\) \(1\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{11}{18}\right)\)
\(\chi_{163}(24,\cdot)\) 163.i 81 Yes \(1\) \(1\) \(e\left(\frac{52}{81}\right)\) \(e\left(\frac{68}{81}\right)\) \(e\left(\frac{23}{81}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{70}{81}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{55}{81}\right)\) \(e\left(\frac{22}{81}\right)\) \(e\left(\frac{14}{81}\right)\)
\(\chi_{163}(25,\cdot)\) 163.g 27 Yes \(1\) \(1\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{19}{27}\right)\)
\(\chi_{163}(26,\cdot)\) 163.i 81 Yes \(1\) \(1\) \(e\left(\frac{26}{81}\right)\) \(e\left(\frac{34}{81}\right)\) \(e\left(\frac{52}{81}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{35}{81}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{68}{81}\right)\) \(e\left(\frac{11}{81}\right)\) \(e\left(\frac{7}{81}\right)\)
\(\chi_{163}(27,\cdot)\) 163.h 54 Yes \(-1\) \(1\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{49}{54}\right)\)
\(\chi_{163}(28,\cdot)\) 163.h 54 Yes \(-1\) \(1\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{41}{54}\right)\)
\(\chi_{163}(29,\cdot)\) 163.j 162 Yes \(-1\) \(1\) \(e\left(\frac{107}{162}\right)\) \(e\left(\frac{115}{162}\right)\) \(e\left(\frac{26}{81}\right)\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{35}{162}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{34}{81}\right)\) \(e\left(\frac{46}{81}\right)\) \(e\left(\frac{7}{162}\right)\)
\(\chi_{163}(30,\cdot)\) 163.f 18 Yes \(-1\) \(1\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{17}{18}\right)\)
\(\chi_{163}(31,\cdot)\) 163.h 54 Yes \(-1\) \(1\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{1}{54}\right)\)
\(\chi_{163}(32,\cdot)\) 163.j 162 Yes \(-1\) \(1\) \(e\left(\frac{5}{162}\right)\) \(e\left(\frac{19}{162}\right)\) \(e\left(\frac{5}{81}\right)\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{41}{162}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{19}{81}\right)\) \(e\left(\frac{40}{81}\right)\) \(e\left(\frac{73}{162}\right)\)