# Properties

 Modulus $199$ Structure $$C_{198}$$ Order $198$

Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(199)

pari: g = idealstar(,199,2)

## Character group

 sage: G.order()  pari: g.no Order = 198 sage: H.invariants()  pari: g.cyc Structure = $$C_{198}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{199}(3,\cdot)$

## First 32 of 198 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{199}(1,\cdot)$$ 199.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{199}(2,\cdot)$$ 199.k 99 yes $$1$$ $$1$$ $$e\left(\frac{74}{99}\right)$$ $$e\left(\frac{53}{99}\right)$$ $$e\left(\frac{49}{99}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{28}{99}\right)$$ $$e\left(\frac{2}{99}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{7}{99}\right)$$ $$e\left(\frac{62}{99}\right)$$ $$e\left(\frac{2}{11}\right)$$
$$\chi_{199}(3,\cdot)$$ 199.l 198 yes $$-1$$ $$1$$ $$e\left(\frac{53}{99}\right)$$ $$e\left(\frac{1}{198}\right)$$ $$e\left(\frac{7}{99}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{107}{198}\right)$$ $$e\left(\frac{71}{99}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{1}{99}\right)$$ $$e\left(\frac{23}{99}\right)$$ $$e\left(\frac{21}{22}\right)$$
$$\chi_{199}(4,\cdot)$$ 199.k 99 yes $$1$$ $$1$$ $$e\left(\frac{49}{99}\right)$$ $$e\left(\frac{7}{99}\right)$$ $$e\left(\frac{98}{99}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{56}{99}\right)$$ $$e\left(\frac{4}{99}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{14}{99}\right)$$ $$e\left(\frac{25}{99}\right)$$ $$e\left(\frac{4}{11}\right)$$
$$\chi_{199}(5,\cdot)$$ 199.i 33 yes $$1$$ $$1$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{8}{11}\right)$$
$$\chi_{199}(6,\cdot)$$ 199.l 198 yes $$-1$$ $$1$$ $$e\left(\frac{28}{99}\right)$$ $$e\left(\frac{107}{198}\right)$$ $$e\left(\frac{56}{99}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{163}{198}\right)$$ $$e\left(\frac{73}{99}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{8}{99}\right)$$ $$e\left(\frac{85}{99}\right)$$ $$e\left(\frac{3}{22}\right)$$
$$\chi_{199}(7,\cdot)$$ 199.k 99 yes $$1$$ $$1$$ $$e\left(\frac{2}{99}\right)$$ $$e\left(\frac{71}{99}\right)$$ $$e\left(\frac{4}{99}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{73}{99}\right)$$ $$e\left(\frac{83}{99}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{43}{99}\right)$$ $$e\left(\frac{98}{99}\right)$$ $$e\left(\frac{6}{11}\right)$$
$$\chi_{199}(8,\cdot)$$ 199.i 33 yes $$1$$ $$1$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{6}{11}\right)$$
$$\chi_{199}(9,\cdot)$$ 199.k 99 yes $$1$$ $$1$$ $$e\left(\frac{7}{99}\right)$$ $$e\left(\frac{1}{99}\right)$$ $$e\left(\frac{14}{99}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{8}{99}\right)$$ $$e\left(\frac{43}{99}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{2}{99}\right)$$ $$e\left(\frac{46}{99}\right)$$ $$e\left(\frac{10}{11}\right)$$
$$\chi_{199}(10,\cdot)$$ 199.k 99 yes $$1$$ $$1$$ $$e\left(\frac{62}{99}\right)$$ $$e\left(\frac{23}{99}\right)$$ $$e\left(\frac{25}{99}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{85}{99}\right)$$ $$e\left(\frac{98}{99}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{46}{99}\right)$$ $$e\left(\frac{68}{99}\right)$$ $$e\left(\frac{10}{11}\right)$$
$$\chi_{199}(11,\cdot)$$ 199.h 22 yes $$-1$$ $$1$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$
$$\chi_{199}(12,\cdot)$$ 199.j 66 yes $$-1$$ $$1$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{5}{66}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{7}{66}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{7}{22}\right)$$
$$\chi_{199}(13,\cdot)$$ 199.k 99 yes $$1$$ $$1$$ $$e\left(\frac{8}{99}\right)$$ $$e\left(\frac{86}{99}\right)$$ $$e\left(\frac{16}{99}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{94}{99}\right)$$ $$e\left(\frac{35}{99}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{73}{99}\right)$$ $$e\left(\frac{95}{99}\right)$$ $$e\left(\frac{2}{11}\right)$$
$$\chi_{199}(14,\cdot)$$ 199.k 99 yes $$1$$ $$1$$ $$e\left(\frac{76}{99}\right)$$ $$e\left(\frac{25}{99}\right)$$ $$e\left(\frac{53}{99}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{2}{99}\right)$$ $$e\left(\frac{85}{99}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{50}{99}\right)$$ $$e\left(\frac{61}{99}\right)$$ $$e\left(\frac{8}{11}\right)$$
$$\chi_{199}(15,\cdot)$$ 199.l 198 yes $$-1$$ $$1$$ $$e\left(\frac{41}{99}\right)$$ $$e\left(\frac{139}{198}\right)$$ $$e\left(\frac{82}{99}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{23}{198}\right)$$ $$e\left(\frac{68}{99}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{40}{99}\right)$$ $$e\left(\frac{29}{99}\right)$$ $$e\left(\frac{15}{22}\right)$$
$$\chi_{199}(16,\cdot)$$ 199.k 99 yes $$1$$ $$1$$ $$e\left(\frac{98}{99}\right)$$ $$e\left(\frac{14}{99}\right)$$ $$e\left(\frac{97}{99}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{13}{99}\right)$$ $$e\left(\frac{8}{99}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{28}{99}\right)$$ $$e\left(\frac{50}{99}\right)$$ $$e\left(\frac{8}{11}\right)$$
$$\chi_{199}(17,\cdot)$$ 199.j 66 yes $$-1$$ $$1$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{41}{66}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{31}{66}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{9}{22}\right)$$
$$\chi_{199}(18,\cdot)$$ 199.f 11 yes $$1$$ $$1$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$
$$\chi_{199}(19,\cdot)$$ 199.g 18 yes $$-1$$ $$1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$-1$$
$$\chi_{199}(20,\cdot)$$ 199.k 99 yes $$1$$ $$1$$ $$e\left(\frac{37}{99}\right)$$ $$e\left(\frac{76}{99}\right)$$ $$e\left(\frac{74}{99}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{14}{99}\right)$$ $$e\left(\frac{1}{99}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{53}{99}\right)$$ $$e\left(\frac{31}{99}\right)$$ $$e\left(\frac{1}{11}\right)$$
$$\chi_{199}(21,\cdot)$$ 199.g 18 yes $$-1$$ $$1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$-1$$
$$\chi_{199}(22,\cdot)$$ 199.l 198 yes $$-1$$ $$1$$ $$e\left(\frac{92}{99}\right)$$ $$e\left(\frac{97}{198}\right)$$ $$e\left(\frac{85}{99}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{83}{198}\right)$$ $$e\left(\frac{56}{99}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{97}{99}\right)$$ $$e\left(\frac{53}{99}\right)$$ $$e\left(\frac{13}{22}\right)$$
$$\chi_{199}(23,\cdot)$$ 199.k 99 yes $$1$$ $$1$$ $$e\left(\frac{17}{99}\right)$$ $$e\left(\frac{59}{99}\right)$$ $$e\left(\frac{34}{99}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{76}{99}\right)$$ $$e\left(\frac{62}{99}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{19}{99}\right)$$ $$e\left(\frac{41}{99}\right)$$ $$e\left(\frac{7}{11}\right)$$
$$\chi_{199}(24,\cdot)$$ 199.g 18 yes $$-1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$-1$$
$$\chi_{199}(25,\cdot)$$ 199.i 33 yes $$1$$ $$1$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{5}{11}\right)$$
$$\chi_{199}(26,\cdot)$$ 199.k 99 yes $$1$$ $$1$$ $$e\left(\frac{82}{99}\right)$$ $$e\left(\frac{40}{99}\right)$$ $$e\left(\frac{65}{99}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{23}{99}\right)$$ $$e\left(\frac{37}{99}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{80}{99}\right)$$ $$e\left(\frac{58}{99}\right)$$ $$e\left(\frac{4}{11}\right)$$
$$\chi_{199}(27,\cdot)$$ 199.j 66 yes $$-1$$ $$1$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{1}{66}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{41}{66}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{19}{22}\right)$$
$$\chi_{199}(28,\cdot)$$ 199.i 33 yes $$1$$ $$1$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{10}{11}\right)$$
$$\chi_{199}(29,\cdot)$$ 199.k 99 yes $$1$$ $$1$$ $$e\left(\frac{47}{99}\right)$$ $$e\left(\frac{35}{99}\right)$$ $$e\left(\frac{94}{99}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{82}{99}\right)$$ $$e\left(\frac{20}{99}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{70}{99}\right)$$ $$e\left(\frac{26}{99}\right)$$ $$e\left(\frac{9}{11}\right)$$
$$\chi_{199}(30,\cdot)$$ 199.l 198 yes $$-1$$ $$1$$ $$e\left(\frac{16}{99}\right)$$ $$e\left(\frac{47}{198}\right)$$ $$e\left(\frac{32}{99}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{79}{198}\right)$$ $$e\left(\frac{70}{99}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{47}{99}\right)$$ $$e\left(\frac{91}{99}\right)$$ $$e\left(\frac{19}{22}\right)$$
$$\chi_{199}(31,\cdot)$$ 199.k 99 yes $$1$$ $$1$$ $$e\left(\frac{79}{99}\right)$$ $$e\left(\frac{82}{99}\right)$$ $$e\left(\frac{59}{99}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{62}{99}\right)$$ $$e\left(\frac{61}{99}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{65}{99}\right)$$ $$e\left(\frac{10}{99}\right)$$ $$e\left(\frac{6}{11}\right)$$
$$\chi_{199}(32,\cdot)$$ 199.k 99 yes $$1$$ $$1$$ $$e\left(\frac{73}{99}\right)$$ $$e\left(\frac{67}{99}\right)$$ $$e\left(\frac{47}{99}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{41}{99}\right)$$ $$e\left(\frac{10}{99}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{35}{99}\right)$$ $$e\left(\frac{13}{99}\right)$$ $$e\left(\frac{10}{11}\right)$$