# Properties

 Modulus $109$ Structure $$C_{108}$$ Order $108$

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 $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$i$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{1}{36}\right)$$