# Properties

 Modulus $155$ Structure $$C_{60}\times C_{2}$$ Order $120$

Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(155)

pari: g = idealstar(,155,2)

## Character group

 sage: G.order()  pari: g.no Order = 120 sage: H.invariants()  pari: g.cyc Structure = $$C_{60}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{155}(32,\cdot)$, $\chi_{155}(96,\cdot)$

## First 32 of 120 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$13$$
$$\chi_{155}(1,\cdot)$$ 155.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{155}(2,\cdot)$$ 155.s 20 yes $$-1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$1$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$
$$\chi_{155}(3,\cdot)$$ 155.x 60 yes $$1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{37}{60}\right)$$
$$\chi_{155}(4,\cdot)$$ 155.n 10 yes $$1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{155}(6,\cdot)$$ 155.k 6 no $$-1$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{155}(7,\cdot)$$ 155.w 60 yes $$-1$$ $$1$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{1}{60}\right)$$
$$\chi_{155}(8,\cdot)$$ 155.s 20 yes $$-1$$ $$1$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$
$$\chi_{155}(9,\cdot)$$ 155.u 30 yes $$1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{7}{30}\right)$$
$$\chi_{155}(11,\cdot)$$ 155.t 30 no $$-1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{13}{30}\right)$$
$$\chi_{155}(12,\cdot)$$ 155.x 60 yes $$1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{43}{60}\right)$$
$$\chi_{155}(13,\cdot)$$ 155.x 60 yes $$1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{60}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{17}{60}\right)$$
$$\chi_{155}(14,\cdot)$$ 155.u 30 yes $$1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{17}{30}\right)$$
$$\chi_{155}(16,\cdot)$$ 155.h 5 no $$1$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{155}(17,\cdot)$$ 155.x 60 yes $$1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{47}{60}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{19}{60}\right)$$
$$\chi_{155}(18,\cdot)$$ 155.w 60 yes $$-1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{60}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{47}{60}\right)$$
$$\chi_{155}(19,\cdot)$$ 155.u 30 yes $$1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{29}{30}\right)$$
$$\chi_{155}(21,\cdot)$$ 155.t 30 no $$-1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{19}{30}\right)$$
$$\chi_{155}(22,\cdot)$$ 155.x 60 yes $$1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{1}{60}\right)$$ $$e\left(\frac{59}{60}\right)$$
$$\chi_{155}(23,\cdot)$$ 155.r 20 yes $$1$$ $$1$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$-1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$
$$\chi_{155}(24,\cdot)$$ 155.v 30 yes $$-1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$
$$\chi_{155}(26,\cdot)$$ 155.k 6 no $$-1$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{155}(27,\cdot)$$ 155.r 20 yes $$1$$ $$1$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$-1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$
$$\chi_{155}(28,\cdot)$$ 155.w 60 yes $$-1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{47}{60}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{7}{60}\right)$$
$$\chi_{155}(29,\cdot)$$ 155.m 10 yes $$-1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$-1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{155}(32,\cdot)$$ 155.g 4 no $$-1$$ $$1$$ $$i$$ $$-i$$ $$-1$$ $$1$$ $$i$$ $$-i$$ $$-1$$ $$1$$ $$i$$ $$-i$$
$$\chi_{155}(33,\cdot)$$ 155.s 20 yes $$-1$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$
$$\chi_{155}(34,\cdot)$$ 155.v 30 yes $$-1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$
$$\chi_{155}(36,\cdot)$$ 155.e 3 no $$1$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{155}(37,\cdot)$$ 155.p 12 yes $$1$$ $$1$$ $$i$$ $$e\left(\frac{7}{12}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{155}(38,\cdot)$$ 155.w 60 yes $$-1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{31}{60}\right)$$
$$\chi_{155}(39,\cdot)$$ 155.n 10 yes $$1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{155}(41,\cdot)$$ 155.q 15 no $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$