# Properties

 Modulus $119$ Structure $$C_{48}\times C_{2}$$ Order $96$

Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(119)

pari: g = idealstar(,119,2)

## Character group

 sage: G.order()  pari: g.no Order = 96 sage: H.invariants()  pari: g.cyc Structure = $$C_{48}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{119}(52,\cdot)$, $\chi_{119}(71,\cdot)$

## 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.

Character Orbit Order Primitive $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$11$$ $$12$$
$$\chi_{119}(1,\cdot)$$ 119.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{119}(2,\cdot)$$ 119.q 24 yes $$1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{24}\right)$$
$$\chi_{119}(3,\cdot)$$ 119.s 48 yes $$1$$ $$1$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{31}{48}\right)$$
$$\chi_{119}(4,\cdot)$$ 119.n 12 yes $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$i$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$
$$\chi_{119}(5,\cdot)$$ 119.s 48 yes $$1$$ $$1$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{11}{48}\right)$$
$$\chi_{119}(6,\cdot)$$ 119.p 16 yes $$1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$
$$\chi_{119}(8,\cdot)$$ 119.k 8 no $$1$$ $$1$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$i$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{119}(9,\cdot)$$ 119.q 24 yes $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{7}{24}\right)$$
$$\chi_{119}(10,\cdot)$$ 119.s 48 yes $$1$$ $$1$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{13}{48}\right)$$
$$\chi_{119}(11,\cdot)$$ 119.t 48 yes $$-1$$ $$1$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{1}{48}\right)$$
$$\chi_{119}(12,\cdot)$$ 119.s 48 yes $$1$$ $$1$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{13}{48}\right)$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{35}{48}\right)$$
$$\chi_{119}(13,\cdot)$$ 119.f 4 yes $$-1$$ $$1$$ $$-1$$ $$-i$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$-1$$ $$i$$ $$-i$$ $$-i$$
$$\chi_{119}(15,\cdot)$$ 119.k 8 no $$1$$ $$1$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{119}(16,\cdot)$$ 119.j 6 yes $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{119}(18,\cdot)$$ 119.e 3 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{119}(19,\cdot)$$ 119.r 24 yes $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{13}{24}\right)$$
$$\chi_{119}(20,\cdot)$$ 119.p 16 yes $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-i$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$
$$\chi_{119}(22,\cdot)$$ 119.o 16 no $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$
$$\chi_{119}(23,\cdot)$$ 119.t 48 yes $$-1$$ $$1$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{13}{48}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{41}{48}\right)$$
$$\chi_{119}(24,\cdot)$$ 119.s 48 yes $$1$$ $$1$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{13}{48}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{37}{48}\right)$$
$$\chi_{119}(25,\cdot)$$ 119.q 24 yes $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{11}{24}\right)$$
$$\chi_{119}(26,\cdot)$$ 119.r 24 yes $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{19}{24}\right)$$
$$\chi_{119}(27,\cdot)$$ 119.p 16 yes $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$
$$\chi_{119}(29,\cdot)$$ 119.o 16 no $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$-i$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$
$$\chi_{119}(30,\cdot)$$ 119.n 12 yes $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$-i$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{119}(31,\cdot)$$ 119.s 48 yes $$1$$ $$1$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{7}{48}\right)$$
$$\chi_{119}(32,\cdot)$$ 119.q 24 yes $$1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{5}{24}\right)$$
$$\chi_{119}(33,\cdot)$$ 119.h 6 yes $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{119}(36,\cdot)$$ 119.k 8 no $$1$$ $$1$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{119}(37,\cdot)$$ 119.t 48 yes $$-1$$ $$1$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{23}{48}\right)$$
$$\chi_{119}(38,\cdot)$$ 119.m 12 yes $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-i$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$
$$\chi_{119}(39,\cdot)$$ 119.t 48 yes $$-1$$ $$1$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{19}{48}\right)$$