Properties

Modulus $109$
Structure \(C_{108}\)
Order $108$

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Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(109)
 
pari: g = idealstar(,109,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 108
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{108}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{109}(6,\cdot)$

First 32 of 108 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.

Character Orbit Order Primitive \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{109}(1,\cdot)\) 109.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{109}(2,\cdot)\) 109.j 36 yes \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{1}{9}\right)\) \(i\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{29}{36}\right)\)
\(\chi_{109}(3,\cdot)\) 109.i 27 yes \(1\) \(1\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{26}{27}\right)\)
\(\chi_{109}(4,\cdot)\) 109.h 18 yes \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(-1\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{11}{18}\right)\)
\(\chi_{109}(5,\cdot)\) 109.i 27 yes \(1\) \(1\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{11}{27}\right)\)
\(\chi_{109}(6,\cdot)\) 109.l 108 yes \(-1\) \(1\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{1}{108}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{25}{108}\right)\) \(e\left(\frac{83}{108}\right)\)
\(\chi_{109}(7,\cdot)\) 109.i 27 yes \(1\) \(1\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{20}{27}\right)\)
\(\chi_{109}(8,\cdot)\) 109.g 12 yes \(-1\) \(1\) \(i\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(-i\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{109}(9,\cdot)\) 109.i 27 yes \(1\) \(1\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{25}{27}\right)\)
\(\chi_{109}(10,\cdot)\) 109.l 108 yes \(-1\) \(1\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{25}{108}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{85}{108}\right)\) \(e\left(\frac{23}{108}\right)\)
\(\chi_{109}(11,\cdot)\) 109.l 108 yes \(-1\) \(1\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{83}{108}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{23}{108}\right)\) \(e\left(\frac{85}{108}\right)\)
\(\chi_{109}(12,\cdot)\) 109.k 54 yes \(1\) \(1\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{31}{54}\right)\)
\(\chi_{109}(13,\cdot)\) 109.l 108 yes \(-1\) \(1\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{67}{108}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{55}{108}\right)\) \(e\left(\frac{53}{108}\right)\)
\(\chi_{109}(14,\cdot)\) 109.l 108 yes \(-1\) \(1\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{97}{108}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{49}{108}\right)\) \(e\left(\frac{59}{108}\right)\)
\(\chi_{109}(15,\cdot)\) 109.i 27 yes \(1\) \(1\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{10}{27}\right)\)
\(\chi_{109}(16,\cdot)\) 109.f 9 yes \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(1\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{9}\right)\)
\(\chi_{109}(17,\cdot)\) 109.j 36 yes \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{4}{9}\right)\) \(i\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{17}{36}\right)\)
\(\chi_{109}(18,\cdot)\) 109.l 108 yes \(-1\) \(1\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{53}{108}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{29}{108}\right)\) \(e\left(\frac{79}{108}\right)\)
\(\chi_{109}(19,\cdot)\) 109.j 36 yes \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{7}{9}\right)\) \(-i\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{23}{36}\right)\)
\(\chi_{109}(20,\cdot)\) 109.k 54 yes \(1\) \(1\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{1}{54}\right)\)
\(\chi_{109}(21,\cdot)\) 109.i 27 yes \(1\) \(1\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{19}{27}\right)\)
\(\chi_{109}(22,\cdot)\) 109.i 27 yes \(1\) \(1\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{16}{27}\right)\)
\(\chi_{109}(23,\cdot)\) 109.j 36 yes \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{2}{9}\right)\) \(i\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{13}{36}\right)\)
\(\chi_{109}(24,\cdot)\) 109.l 108 yes \(-1\) \(1\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{7}{108}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{67}{108}\right)\) \(e\left(\frac{41}{108}\right)\)
\(\chi_{109}(25,\cdot)\) 109.i 27 yes \(1\) \(1\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{22}{27}\right)\)
\(\chi_{109}(26,\cdot)\) 109.i 27 yes \(1\) \(1\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{8}{27}\right)\)
\(\chi_{109}(27,\cdot)\) 109.f 9 yes \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(1\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{109}(28,\cdot)\) 109.k 54 yes \(1\) \(1\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{19}{54}\right)\)
\(\chi_{109}(29,\cdot)\) 109.k 54 yes \(1\) \(1\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{7}{54}\right)\)
\(\chi_{109}(30,\cdot)\) 109.l 108 yes \(-1\) \(1\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{77}{108}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{89}{108}\right)\) \(e\left(\frac{19}{108}\right)\)
\(\chi_{109}(31,\cdot)\) 109.k 54 yes \(1\) \(1\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{41}{54}\right)\)
\(\chi_{109}(32,\cdot)\) 109.j 36 yes \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{5}{9}\right)\) \(i\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{1}{36}\right)\)