# Properties

 Modulus $153$ Structure $$C_{2}\times C_{48}$$ Order $96$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(153)

pari: g = idealstar(,153,2)

## Character group

 sage: G.order()  pari: g.no Order = 96 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{48}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{153}(137,\cdot)$, $\chi_{153}(37,\cdot)$

## 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{153}(1,\cdot)$$ 153.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{153}(2,\cdot)$$ 153.q 24 yes $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{153}(4,\cdot)$$ 153.n 12 yes $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-1$$ $$i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{153}(5,\cdot)$$ 153.s 48 yes $$1$$ $$1$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{153}(7,\cdot)$$ 153.t 48 yes $$-1$$ $$1$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{153}(8,\cdot)$$ 153.k 8 no $$-1$$ $$1$$ $$i$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$1$$
$$\chi_{153}(10,\cdot)$$ 153.p 16 no $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$e\left(\frac{11}{16}\right)$$ $$-1$$
$$\chi_{153}(11,\cdot)$$ 153.s 48 yes $$1$$ $$1$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{153}(13,\cdot)$$ 153.n 12 yes $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$-1$$ $$-i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{153}(14,\cdot)$$ 153.s 48 yes $$1$$ $$1$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{153}(16,\cdot)$$ 153.h 6 yes $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{153}(19,\cdot)$$ 153.l 8 no $$1$$ $$1$$ $$i$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$1$$
$$\chi_{153}(20,\cdot)$$ 153.s 48 yes $$1$$ $$1$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{153}(22,\cdot)$$ 153.t 48 yes $$-1$$ $$1$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{153}(23,\cdot)$$ 153.s 48 yes $$1$$ $$1$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{153}(25,\cdot)$$ 153.r 24 yes $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{153}(26,\cdot)$$ 153.k 8 no $$-1$$ $$1$$ $$i$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$ $$1$$
$$\chi_{153}(28,\cdot)$$ 153.p 16 no $$-1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-i$$ $$e\left(\frac{15}{16}\right)$$ $$-1$$
$$\chi_{153}(29,\cdot)$$ 153.s 48 yes $$1$$ $$1$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{153}(31,\cdot)$$ 153.t 48 yes $$-1$$ $$1$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{48}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{153}(32,\cdot)$$ 153.q 24 yes $$-1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{153}(35,\cdot)$$ 153.b 2 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$
$$\chi_{153}(37,\cdot)$$ 153.p 16 no $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$i$$ $$e\left(\frac{9}{16}\right)$$ $$-1$$
$$\chi_{153}(38,\cdot)$$ 153.m 12 yes $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$1$$ $$i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{153}(40,\cdot)$$ 153.t 48 yes $$-1$$ $$1$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{153}(41,\cdot)$$ 153.s 48 yes $$1$$ $$1$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{153}(43,\cdot)$$ 153.r 24 yes $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{153}(44,\cdot)$$ 153.o 16 no $$1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$-i$$ $$e\left(\frac{3}{16}\right)$$ $$-1$$
$$\chi_{153}(46,\cdot)$$ 153.p 16 no $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$i$$ $$e\left(\frac{5}{16}\right)$$ $$-1$$
$$\chi_{153}(47,\cdot)$$ 153.m 12 yes $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$1$$ $$-i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{153}(49,\cdot)$$ 153.r 24 yes $$1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{153}(50,\cdot)$$ 153.i 6 yes $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$