# Properties

 Modulus $847$ Structure $$C_{2}\times C_{330}$$ Order $660$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(847)

pari: g = idealstar(,847,2)

## Character group

 sage: G.order()  pari: g.no Order = 660 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{330}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{847}(122,\cdot)$, $\chi_{847}(365,\cdot)$

## First 32 of 660 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$12$$ $$13$$
$$\chi_{847}(1,\cdot)$$ 847.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{847}(2,\cdot)$$ 847.bf 330 yes $$-1$$ $$1$$ $$e\left(\frac{223}{330}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{58}{165}\right)$$ $$e\left(\frac{56}{165}\right)$$ $$e\left(\frac{89}{110}\right)$$ $$e\left(\frac{3}{110}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{66}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{101}{110}\right)$$
$$\chi_{847}(3,\cdot)$$ 847.t 30 no $$-1$$ $$1$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{847}(4,\cdot)$$ 847.bc 165 yes $$1$$ $$1$$ $$e\left(\frac{58}{165}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{116}{165}\right)$$ $$e\left(\frac{112}{165}\right)$$ $$e\left(\frac{34}{55}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{46}{55}\right)$$
$$\chi_{847}(5,\cdot)$$ 847.bd 330 yes $$-1$$ $$1$$ $$e\left(\frac{56}{165}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{112}{165}\right)$$ $$e\left(\frac{313}{330}\right)$$ $$e\left(\frac{41}{110}\right)$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{19}{66}\right)$$ $$e\left(\frac{47}{66}\right)$$ $$e\left(\frac{49}{110}\right)$$
$$\chi_{847}(6,\cdot)$$ 847.ba 110 yes $$1$$ $$1$$ $$e\left(\frac{89}{110}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{34}{55}\right)$$ $$e\left(\frac{41}{110}\right)$$ $$e\left(\frac{28}{55}\right)$$ $$e\left(\frac{47}{110}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{12}{55}\right)$$
$$\chi_{847}(8,\cdot)$$ 847.z 110 no $$-1$$ $$1$$ $$e\left(\frac{3}{110}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{47}{110}\right)$$ $$e\left(\frac{9}{110}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{83}{110}\right)$$
$$\chi_{847}(9,\cdot)$$ 847.n 15 no $$1$$ $$1$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{847}(10,\cdot)$$ 847.x 66 yes $$1$$ $$1$$ $$e\left(\frac{1}{66}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{19}{66}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{13}{66}\right)$$ $$e\left(\frac{4}{11}\right)$$
$$\chi_{847}(12,\cdot)$$ 847.y 66 yes $$-1$$ $$1$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{47}{66}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{13}{66}\right)$$ $$e\left(\frac{53}{66}\right)$$ $$e\left(\frac{3}{22}\right)$$
$$\chi_{847}(13,\cdot)$$ 847.ba 110 yes $$1$$ $$1$$ $$e\left(\frac{101}{110}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{49}{110}\right)$$ $$e\left(\frac{12}{55}\right)$$ $$e\left(\frac{83}{110}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{13}{55}\right)$$
$$\chi_{847}(15,\cdot)$$ 847.v 55 no $$1$$ $$1$$ $$e\left(\frac{26}{55}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{52}{55}\right)$$ $$e\left(\frac{54}{55}\right)$$ $$e\left(\frac{4}{55}\right)$$ $$e\left(\frac{23}{55}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{41}{55}\right)$$
$$\chi_{847}(16,\cdot)$$ 847.bc 165 yes $$1$$ $$1$$ $$e\left(\frac{116}{165}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{67}{165}\right)$$ $$e\left(\frac{59}{165}\right)$$ $$e\left(\frac{13}{55}\right)$$ $$e\left(\frac{6}{55}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{37}{55}\right)$$
$$\chi_{847}(17,\cdot)$$ 847.be 330 yes $$1$$ $$1$$ $$e\left(\frac{257}{330}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{92}{165}\right)$$ $$e\left(\frac{263}{330}\right)$$ $$e\left(\frac{8}{55}\right)$$ $$e\left(\frac{37}{110}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{61}{66}\right)$$ $$e\left(\frac{27}{55}\right)$$
$$\chi_{847}(18,\cdot)$$ 847.bf 330 yes $$-1$$ $$1$$ $$e\left(\frac{311}{330}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{146}{165}\right)$$ $$e\left(\frac{67}{165}\right)$$ $$e\left(\frac{23}{110}\right)$$ $$e\left(\frac{91}{110}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{23}{66}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{57}{110}\right)$$
$$\chi_{847}(19,\cdot)$$ 847.be 330 yes $$1$$ $$1$$ $$e\left(\frac{139}{330}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{139}{165}\right)$$ $$e\left(\frac{1}{330}\right)$$ $$e\left(\frac{36}{55}\right)$$ $$e\left(\frac{29}{110}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{5}{66}\right)$$ $$e\left(\frac{39}{55}\right)$$
$$\chi_{847}(20,\cdot)$$ 847.bb 110 yes $$-1$$ $$1$$ $$e\left(\frac{38}{55}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{21}{55}\right)$$ $$e\left(\frac{69}{110}\right)$$ $$e\left(\frac{109}{110}\right)$$ $$e\left(\frac{4}{55}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{31}{110}\right)$$
$$\chi_{847}(23,\cdot)$$ 847.u 33 yes $$1$$ $$1$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{3}{11}\right)$$
$$\chi_{847}(24,\cdot)$$ 847.be 330 yes $$1$$ $$1$$ $$e\left(\frac{53}{330}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{53}{165}\right)$$ $$e\left(\frac{17}{330}\right)$$ $$e\left(\frac{7}{55}\right)$$ $$e\left(\frac{53}{110}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{19}{66}\right)$$ $$e\left(\frac{3}{55}\right)$$
$$\chi_{847}(25,\cdot)$$ 847.bc 165 yes $$1$$ $$1$$ $$e\left(\frac{112}{165}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{59}{165}\right)$$ $$e\left(\frac{148}{165}\right)$$ $$e\left(\frac{41}{55}\right)$$ $$e\left(\frac{2}{55}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{49}{55}\right)$$
$$\chi_{847}(26,\cdot)$$ 847.bd 330 yes $$-1$$ $$1$$ $$e\left(\frac{98}{165}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{31}{165}\right)$$ $$e\left(\frac{259}{330}\right)$$ $$e\left(\frac{3}{110}\right)$$ $$e\left(\frac{43}{55}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{25}{66}\right)$$ $$e\left(\frac{41}{66}\right)$$ $$e\left(\frac{17}{110}\right)$$
$$\chi_{847}(27,\cdot)$$ 847.j 10 no $$-1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{847}(29,\cdot)$$ 847.z 110 no $$-1$$ $$1$$ $$e\left(\frac{17}{110}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{17}{55}\right)$$ $$e\left(\frac{24}{55}\right)$$ $$e\left(\frac{83}{110}\right)$$ $$e\left(\frac{51}{110}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{67}{110}\right)$$
$$\chi_{847}(30,\cdot)$$ 847.bf 330 yes $$-1$$ $$1$$ $$e\left(\frac{49}{330}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{49}{165}\right)$$ $$e\left(\frac{53}{165}\right)$$ $$e\left(\frac{97}{110}\right)$$ $$e\left(\frac{49}{110}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{31}{66}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{73}{110}\right)$$
$$\chi_{847}(31,\cdot)$$ 847.bd 330 yes $$-1$$ $$1$$ $$e\left(\frac{19}{165}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{38}{165}\right)$$ $$e\left(\frac{227}{330}\right)$$ $$e\left(\frac{9}{110}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{53}{66}\right)$$ $$e\left(\frac{13}{66}\right)$$ $$e\left(\frac{51}{110}\right)$$
$$\chi_{847}(32,\cdot)$$ 847.w 66 yes $$-1$$ $$1$$ $$e\left(\frac{25}{66}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{66}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{13}{22}\right)$$
$$\chi_{847}(34,\cdot)$$ 847.o 22 yes $$-1$$ $$1$$ $$e\left(\frac{5}{11}\right)$$ $$-1$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$1$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$
$$\chi_{847}(36,\cdot)$$ 847.v 55 no $$1$$ $$1$$ $$e\left(\frac{34}{55}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{13}{55}\right)$$ $$e\left(\frac{41}{55}\right)$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{47}{55}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{24}{55}\right)$$
$$\chi_{847}(37,\cdot)$$ 847.bc 165 yes $$1$$ $$1$$ $$e\left(\frac{8}{165}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{16}{165}\right)$$ $$e\left(\frac{152}{165}\right)$$ $$e\left(\frac{54}{55}\right)$$ $$e\left(\frac{8}{55}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{31}{55}\right)$$
$$\chi_{847}(38,\cdot)$$ 847.bd 330 yes $$-1$$ $$1$$ $$e\left(\frac{16}{165}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{32}{165}\right)$$ $$e\left(\frac{113}{330}\right)$$ $$e\left(\frac{51}{110}\right)$$ $$e\left(\frac{16}{55}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{29}{66}\right)$$ $$e\left(\frac{37}{66}\right)$$ $$e\left(\frac{69}{110}\right)$$
$$\chi_{847}(39,\cdot)$$ 847.bf 330 yes $$-1$$ $$1$$ $$e\left(\frac{17}{330}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{17}{165}\right)$$ $$e\left(\frac{79}{165}\right)$$ $$e\left(\frac{101}{110}\right)$$ $$e\left(\frac{17}{110}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{35}{66}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{59}{110}\right)$$
$$\chi_{847}(40,\cdot)$$ 847.r 30 no $$1$$ $$1$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{5}\right)$$