# Properties

 Modulus $363$ Structure $$C_{2}\times C_{110}$$ Order $220$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(363)

pari: g = idealstar(,363,2)

## Character group

 sage: G.order()  pari: g.no Order = 220 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{110}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{363}(122,\cdot)$, $\chi_{363}(244,\cdot)$

## First 32 of 220 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$13$$ $$14$$ $$16$$ $$17$$
$$\chi_{363}(1,\cdot)$$ 363.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{363}(2,\cdot)$$ 363.p 110 yes $$1$$ $$1$$ $$e\left(\frac{28}{55}\right)$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{19}{110}\right)$$ $$e\left(\frac{7}{110}\right)$$ $$e\left(\frac{29}{55}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{101}{110}\right)$$ $$e\left(\frac{63}{110}\right)$$ $$e\left(\frac{2}{55}\right)$$ $$e\left(\frac{52}{55}\right)$$
$$\chi_{363}(4,\cdot)$$ 363.m 55 no $$1$$ $$1$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{2}{55}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{7}{55}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{8}{55}\right)$$ $$e\left(\frac{4}{55}\right)$$ $$e\left(\frac{49}{55}\right)$$
$$\chi_{363}(5,\cdot)$$ 363.n 110 yes $$-1$$ $$1$$ $$e\left(\frac{19}{110}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{31}{110}\right)$$ $$e\left(\frac{39}{55}\right)$$ $$e\left(\frac{57}{110}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{52}{55}\right)$$ $$e\left(\frac{97}{110}\right)$$ $$e\left(\frac{38}{55}\right)$$ $$e\left(\frac{51}{110}\right)$$
$$\chi_{363}(7,\cdot)$$ 363.o 110 no $$-1$$ $$1$$ $$e\left(\frac{7}{110}\right)$$ $$e\left(\frac{7}{55}\right)$$ $$e\left(\frac{39}{55}\right)$$ $$e\left(\frac{49}{110}\right)$$ $$e\left(\frac{21}{110}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{47}{110}\right)$$ $$e\left(\frac{28}{55}\right)$$ $$e\left(\frac{14}{55}\right)$$ $$e\left(\frac{13}{110}\right)$$
$$\chi_{363}(8,\cdot)$$ 363.p 110 yes $$1$$ $$1$$ $$e\left(\frac{29}{55}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{57}{110}\right)$$ $$e\left(\frac{21}{110}\right)$$ $$e\left(\frac{32}{55}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{83}{110}\right)$$ $$e\left(\frac{79}{110}\right)$$ $$e\left(\frac{6}{55}\right)$$ $$e\left(\frac{46}{55}\right)$$
$$\chi_{363}(10,\cdot)$$ 363.k 22 no $$-1$$ $$1$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$
$$\chi_{363}(13,\cdot)$$ 363.o 110 no $$-1$$ $$1$$ $$e\left(\frac{101}{110}\right)$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{52}{55}\right)$$ $$e\left(\frac{47}{110}\right)$$ $$e\left(\frac{83}{110}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{81}{110}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{37}{55}\right)$$ $$e\left(\frac{109}{110}\right)$$
$$\chi_{363}(14,\cdot)$$ 363.n 110 yes $$-1$$ $$1$$ $$e\left(\frac{63}{110}\right)$$ $$e\left(\frac{8}{55}\right)$$ $$e\left(\frac{97}{110}\right)$$ $$e\left(\frac{28}{55}\right)$$ $$e\left(\frac{79}{110}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{9}{110}\right)$$ $$e\left(\frac{16}{55}\right)$$ $$e\left(\frac{7}{110}\right)$$
$$\chi_{363}(16,\cdot)$$ 363.m 55 no $$1$$ $$1$$ $$e\left(\frac{2}{55}\right)$$ $$e\left(\frac{4}{55}\right)$$ $$e\left(\frac{38}{55}\right)$$ $$e\left(\frac{14}{55}\right)$$ $$e\left(\frac{6}{55}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{37}{55}\right)$$ $$e\left(\frac{16}{55}\right)$$ $$e\left(\frac{8}{55}\right)$$ $$e\left(\frac{43}{55}\right)$$
$$\chi_{363}(17,\cdot)$$ 363.p 110 yes $$1$$ $$1$$ $$e\left(\frac{52}{55}\right)$$ $$e\left(\frac{49}{55}\right)$$ $$e\left(\frac{51}{110}\right)$$ $$e\left(\frac{13}{110}\right)$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{109}{110}\right)$$ $$e\left(\frac{7}{110}\right)$$ $$e\left(\frac{43}{55}\right)$$ $$e\left(\frac{18}{55}\right)$$
$$\chi_{363}(19,\cdot)$$ 363.o 110 no $$-1$$ $$1$$ $$e\left(\frac{83}{110}\right)$$ $$e\left(\frac{28}{55}\right)$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{31}{110}\right)$$ $$e\left(\frac{29}{110}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{23}{110}\right)$$ $$e\left(\frac{2}{55}\right)$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{107}{110}\right)$$
$$\chi_{363}(20,\cdot)$$ 363.n 110 yes $$-1$$ $$1$$ $$e\left(\frac{21}{110}\right)$$ $$e\left(\frac{21}{55}\right)$$ $$e\left(\frac{69}{110}\right)$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{63}{110}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{43}{55}\right)$$ $$e\left(\frac{3}{110}\right)$$ $$e\left(\frac{42}{55}\right)$$ $$e\left(\frac{39}{110}\right)$$
$$\chi_{363}(23,\cdot)$$ 363.l 22 yes $$-1$$ $$1$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$
$$\chi_{363}(25,\cdot)$$ 363.m 55 no $$1$$ $$1$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{38}{55}\right)$$ $$e\left(\frac{31}{55}\right)$$ $$e\left(\frac{23}{55}\right)$$ $$e\left(\frac{2}{55}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{49}{55}\right)$$ $$e\left(\frac{42}{55}\right)$$ $$e\left(\frac{21}{55}\right)$$ $$e\left(\frac{51}{55}\right)$$
$$\chi_{363}(26,\cdot)$$ 363.n 110 yes $$-1$$ $$1$$ $$e\left(\frac{47}{110}\right)$$ $$e\left(\frac{47}{55}\right)$$ $$e\left(\frac{13}{110}\right)$$ $$e\left(\frac{27}{55}\right)$$ $$e\left(\frac{31}{110}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{36}{55}\right)$$ $$e\left(\frac{101}{110}\right)$$ $$e\left(\frac{39}{55}\right)$$ $$e\left(\frac{103}{110}\right)$$
$$\chi_{363}(28,\cdot)$$ 363.o 110 no $$-1$$ $$1$$ $$e\left(\frac{9}{110}\right)$$ $$e\left(\frac{9}{55}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{63}{110}\right)$$ $$e\left(\frac{27}{110}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{29}{110}\right)$$ $$e\left(\frac{36}{55}\right)$$ $$e\left(\frac{18}{55}\right)$$ $$e\left(\frac{1}{110}\right)$$
$$\chi_{363}(29,\cdot)$$ 363.p 110 yes $$1$$ $$1$$ $$e\left(\frac{36}{55}\right)$$ $$e\left(\frac{17}{55}\right)$$ $$e\left(\frac{103}{110}\right)$$ $$e\left(\frac{9}{110}\right)$$ $$e\left(\frac{53}{55}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{67}{110}\right)$$ $$e\left(\frac{81}{110}\right)$$ $$e\left(\frac{34}{55}\right)$$ $$e\left(\frac{4}{55}\right)$$
$$\chi_{363}(31,\cdot)$$ 363.m 55 no $$1$$ $$1$$ $$e\left(\frac{43}{55}\right)$$ $$e\left(\frac{31}{55}\right)$$ $$e\left(\frac{47}{55}\right)$$ $$e\left(\frac{26}{55}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{53}{55}\right)$$ $$e\left(\frac{14}{55}\right)$$ $$e\left(\frac{7}{55}\right)$$ $$e\left(\frac{17}{55}\right)$$
$$\chi_{363}(32,\cdot)$$ 363.j 22 yes $$1$$ $$1$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$
$$\chi_{363}(34,\cdot)$$ 363.i 11 no $$1$$ $$1$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$
$$\chi_{363}(35,\cdot)$$ 363.p 110 yes $$1$$ $$1$$ $$e\left(\frac{13}{55}\right)$$ $$e\left(\frac{26}{55}\right)$$ $$e\left(\frac{109}{110}\right)$$ $$e\left(\frac{17}{110}\right)$$ $$e\left(\frac{39}{55}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{41}{110}\right)$$ $$e\left(\frac{43}{110}\right)$$ $$e\left(\frac{52}{55}\right)$$ $$e\left(\frac{32}{55}\right)$$
$$\chi_{363}(37,\cdot)$$ 363.m 55 no $$1$$ $$1$$ $$e\left(\frac{21}{55}\right)$$ $$e\left(\frac{42}{55}\right)$$ $$e\left(\frac{14}{55}\right)$$ $$e\left(\frac{37}{55}\right)$$ $$e\left(\frac{8}{55}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{31}{55}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{29}{55}\right)$$ $$e\left(\frac{39}{55}\right)$$
$$\chi_{363}(38,\cdot)$$ 363.n 110 yes $$-1$$ $$1$$ $$e\left(\frac{29}{110}\right)$$ $$e\left(\frac{29}{55}\right)$$ $$e\left(\frac{1}{110}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{87}{110}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{7}{55}\right)$$ $$e\left(\frac{67}{110}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{101}{110}\right)$$
$$\chi_{363}(40,\cdot)$$ 363.g 10 no $$-1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$-1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{363}(41,\cdot)$$ 363.p 110 yes $$1$$ $$1$$ $$e\left(\frac{39}{55}\right)$$ $$e\left(\frac{23}{55}\right)$$ $$e\left(\frac{107}{110}\right)$$ $$e\left(\frac{51}{110}\right)$$ $$e\left(\frac{7}{55}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{13}{110}\right)$$ $$e\left(\frac{19}{110}\right)$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{41}{55}\right)$$
$$\chi_{363}(43,\cdot)$$ 363.k 22 no $$-1$$ $$1$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$
$$\chi_{363}(46,\cdot)$$ 363.o 110 no $$-1$$ $$1$$ $$e\left(\frac{71}{110}\right)$$ $$e\left(\frac{16}{55}\right)$$ $$e\left(\frac{42}{55}\right)$$ $$e\left(\frac{57}{110}\right)$$ $$e\left(\frac{103}{110}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{21}{110}\right)$$ $$e\left(\frac{9}{55}\right)$$ $$e\left(\frac{32}{55}\right)$$ $$e\left(\frac{69}{110}\right)$$
$$\chi_{363}(47,\cdot)$$ 363.n 110 yes $$-1$$ $$1$$ $$e\left(\frac{43}{110}\right)$$ $$e\left(\frac{43}{55}\right)$$ $$e\left(\frac{47}{110}\right)$$ $$e\left(\frac{13}{55}\right)$$ $$e\left(\frac{19}{110}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{54}{55}\right)$$ $$e\left(\frac{69}{110}\right)$$ $$e\left(\frac{31}{55}\right)$$ $$e\left(\frac{17}{110}\right)$$
$$\chi_{363}(49,\cdot)$$ 363.m 55 no $$1$$ $$1$$ $$e\left(\frac{7}{55}\right)$$ $$e\left(\frac{14}{55}\right)$$ $$e\left(\frac{23}{55}\right)$$ $$e\left(\frac{49}{55}\right)$$ $$e\left(\frac{21}{55}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{47}{55}\right)$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{28}{55}\right)$$ $$e\left(\frac{13}{55}\right)$$
$$\chi_{363}(50,\cdot)$$ 363.p 110 yes $$1$$ $$1$$ $$e\left(\frac{47}{55}\right)$$ $$e\left(\frac{39}{55}\right)$$ $$e\left(\frac{81}{110}\right)$$ $$e\left(\frac{53}{110}\right)$$ $$e\left(\frac{31}{55}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{89}{110}\right)$$ $$e\left(\frac{37}{110}\right)$$ $$e\left(\frac{23}{55}\right)$$ $$e\left(\frac{48}{55}\right)$$
$$\chi_{363}(52,\cdot)$$ 363.o 110 no $$-1$$ $$1$$ $$e\left(\frac{103}{110}\right)$$ $$e\left(\frac{48}{55}\right)$$ $$e\left(\frac{16}{55}\right)$$ $$e\left(\frac{61}{110}\right)$$ $$e\left(\frac{89}{110}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{63}{110}\right)$$ $$e\left(\frac{27}{55}\right)$$ $$e\left(\frac{41}{55}\right)$$ $$e\left(\frac{97}{110}\right)$$