# Properties

 Modulus $1089$ Structure $$C_{330}\times C_{2}$$ Order $660$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(1089)

pari: g = idealstar(,1089,2)

## Character group

 sage: G.order()  pari: g.no Order = 660 sage: H.invariants()  pari: g.cyc Structure = $$C_{330}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{1089}(848,\cdot)$, $\chi_{1089}(244,\cdot)$

## 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$13$$ $$14$$ $$16$$ $$17$$
$$\chi_{1089}(1,\cdot)$$ 1089.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{1089}(2,\cdot)$$ 1089.bd 330 yes $$1$$ $$1$$ $$e\left(\frac{29}{165}\right)$$ $$e\left(\frac{58}{165}\right)$$ $$e\left(\frac{167}{330}\right)$$ $$e\left(\frac{241}{330}\right)$$ $$e\left(\frac{29}{55}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{83}{330}\right)$$ $$e\left(\frac{299}{330}\right)$$ $$e\left(\frac{116}{165}\right)$$ $$e\left(\frac{52}{55}\right)$$
$$\chi_{1089}(4,\cdot)$$ 1089.bc 165 yes $$1$$ $$1$$ $$e\left(\frac{58}{165}\right)$$ $$e\left(\frac{116}{165}\right)$$ $$e\left(\frac{2}{165}\right)$$ $$e\left(\frac{76}{165}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{83}{165}\right)$$ $$e\left(\frac{134}{165}\right)$$ $$e\left(\frac{67}{165}\right)$$ $$e\left(\frac{49}{55}\right)$$
$$\chi_{1089}(5,\cdot)$$ 1089.bf 330 yes $$-1$$ $$1$$ $$e\left(\frac{167}{330}\right)$$ $$e\left(\frac{2}{165}\right)$$ $$e\left(\frac{313}{330}\right)$$ $$e\left(\frac{7}{165}\right)$$ $$e\left(\frac{57}{110}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{101}{165}\right)$$ $$e\left(\frac{181}{330}\right)$$ $$e\left(\frac{4}{165}\right)$$ $$e\left(\frac{51}{110}\right)$$
$$\chi_{1089}(7,\cdot)$$ 1089.be 330 yes $$-1$$ $$1$$ $$e\left(\frac{241}{330}\right)$$ $$e\left(\frac{76}{165}\right)$$ $$e\left(\frac{7}{165}\right)$$ $$e\left(\frac{37}{330}\right)$$ $$e\left(\frac{21}{110}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{251}{330}\right)$$ $$e\left(\frac{139}{165}\right)$$ $$e\left(\frac{152}{165}\right)$$ $$e\left(\frac{13}{110}\right)$$
$$\chi_{1089}(8,\cdot)$$ 1089.bb 110 no $$1$$ $$1$$ $$e\left(\frac{29}{55}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{57}{110}\right)$$ $$e\left(\frac{21}{110}\right)$$ $$e\left(\frac{32}{55}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{83}{110}\right)$$ $$e\left(\frac{79}{110}\right)$$ $$e\left(\frac{6}{55}\right)$$ $$e\left(\frac{46}{55}\right)$$
$$\chi_{1089}(10,\cdot)$$ 1089.p 22 no $$-1$$ $$1$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$
$$\chi_{1089}(13,\cdot)$$ 1089.be 330 yes $$-1$$ $$1$$ $$e\left(\frac{83}{330}\right)$$ $$e\left(\frac{83}{165}\right)$$ $$e\left(\frac{101}{165}\right)$$ $$e\left(\frac{251}{330}\right)$$ $$e\left(\frac{83}{110}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{133}{330}\right)$$ $$e\left(\frac{2}{165}\right)$$ $$e\left(\frac{1}{165}\right)$$ $$e\left(\frac{109}{110}\right)$$
$$\chi_{1089}(14,\cdot)$$ 1089.bf 330 yes $$-1$$ $$1$$ $$e\left(\frac{299}{330}\right)$$ $$e\left(\frac{134}{165}\right)$$ $$e\left(\frac{181}{330}\right)$$ $$e\left(\frac{139}{165}\right)$$ $$e\left(\frac{79}{110}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{2}{165}\right)$$ $$e\left(\frac{247}{330}\right)$$ $$e\left(\frac{103}{165}\right)$$ $$e\left(\frac{7}{110}\right)$$
$$\chi_{1089}(16,\cdot)$$ 1089.bc 165 yes $$1$$ $$1$$ $$e\left(\frac{116}{165}\right)$$ $$e\left(\frac{67}{165}\right)$$ $$e\left(\frac{4}{165}\right)$$ $$e\left(\frac{152}{165}\right)$$ $$e\left(\frac{6}{55}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{1}{165}\right)$$ $$e\left(\frac{103}{165}\right)$$ $$e\left(\frac{134}{165}\right)$$ $$e\left(\frac{43}{55}\right)$$
$$\chi_{1089}(17,\cdot)$$ 1089.bb 110 no $$1$$ $$1$$ $$e\left(\frac{52}{55}\right)$$ $$e\left(\frac{49}{55}\right)$$ $$e\left(\frac{51}{110}\right)$$ $$e\left(\frac{13}{110}\right)$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{109}{110}\right)$$ $$e\left(\frac{7}{110}\right)$$ $$e\left(\frac{43}{55}\right)$$ $$e\left(\frac{18}{55}\right)$$
$$\chi_{1089}(19,\cdot)$$ 1089.ba 110 no $$-1$$ $$1$$ $$e\left(\frac{83}{110}\right)$$ $$e\left(\frac{28}{55}\right)$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{31}{110}\right)$$ $$e\left(\frac{29}{110}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{23}{110}\right)$$ $$e\left(\frac{2}{55}\right)$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{107}{110}\right)$$
$$\chi_{1089}(20,\cdot)$$ 1089.bf 330 yes $$-1$$ $$1$$ $$e\left(\frac{283}{330}\right)$$ $$e\left(\frac{118}{165}\right)$$ $$e\left(\frac{317}{330}\right)$$ $$e\left(\frac{83}{165}\right)$$ $$e\left(\frac{63}{110}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{19}{165}\right)$$ $$e\left(\frac{119}{330}\right)$$ $$e\left(\frac{71}{165}\right)$$ $$e\left(\frac{39}{110}\right)$$
$$\chi_{1089}(23,\cdot)$$ 1089.w 66 yes $$-1$$ $$1$$ $$e\left(\frac{31}{66}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{17}{66}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{17}{66}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{15}{22}\right)$$
$$\chi_{1089}(25,\cdot)$$ 1089.bc 165 yes $$1$$ $$1$$ $$e\left(\frac{2}{165}\right)$$ $$e\left(\frac{4}{165}\right)$$ $$e\left(\frac{148}{165}\right)$$ $$e\left(\frac{14}{165}\right)$$ $$e\left(\frac{2}{55}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{37}{165}\right)$$ $$e\left(\frac{16}{165}\right)$$ $$e\left(\frac{8}{165}\right)$$ $$e\left(\frac{51}{55}\right)$$
$$\chi_{1089}(26,\cdot)$$ 1089.z 110 no $$-1$$ $$1$$ $$e\left(\frac{47}{110}\right)$$ $$e\left(\frac{47}{55}\right)$$ $$e\left(\frac{13}{110}\right)$$ $$e\left(\frac{27}{55}\right)$$ $$e\left(\frac{31}{110}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{36}{55}\right)$$ $$e\left(\frac{101}{110}\right)$$ $$e\left(\frac{39}{55}\right)$$ $$e\left(\frac{103}{110}\right)$$
$$\chi_{1089}(28,\cdot)$$ 1089.ba 110 no $$-1$$ $$1$$ $$e\left(\frac{9}{110}\right)$$ $$e\left(\frac{9}{55}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{63}{110}\right)$$ $$e\left(\frac{27}{110}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{29}{110}\right)$$ $$e\left(\frac{36}{55}\right)$$ $$e\left(\frac{18}{55}\right)$$ $$e\left(\frac{1}{110}\right)$$
$$\chi_{1089}(29,\cdot)$$ 1089.bd 330 yes $$1$$ $$1$$ $$e\left(\frac{53}{165}\right)$$ $$e\left(\frac{106}{165}\right)$$ $$e\left(\frac{89}{330}\right)$$ $$e\left(\frac{247}{330}\right)$$ $$e\left(\frac{53}{55}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{311}{330}\right)$$ $$e\left(\frac{23}{330}\right)$$ $$e\left(\frac{47}{165}\right)$$ $$e\left(\frac{4}{55}\right)$$
$$\chi_{1089}(31,\cdot)$$ 1089.bc 165 yes $$1$$ $$1$$ $$e\left(\frac{19}{165}\right)$$ $$e\left(\frac{38}{165}\right)$$ $$e\left(\frac{86}{165}\right)$$ $$e\left(\frac{133}{165}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{104}{165}\right)$$ $$e\left(\frac{152}{165}\right)$$ $$e\left(\frac{76}{165}\right)$$ $$e\left(\frac{17}{55}\right)$$
$$\chi_{1089}(32,\cdot)$$ 1089.y 66 yes $$1$$ $$1$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{35}{66}\right)$$ $$e\left(\frac{43}{66}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{17}{66}\right)$$ $$e\left(\frac{35}{66}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{8}{11}\right)$$
$$\chi_{1089}(34,\cdot)$$ 1089.u 33 yes $$1$$ $$1$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{3}{11}\right)$$
$$\chi_{1089}(35,\cdot)$$ 1089.bb 110 no $$1$$ $$1$$ $$e\left(\frac{13}{55}\right)$$ $$e\left(\frac{26}{55}\right)$$ $$e\left(\frac{109}{110}\right)$$ $$e\left(\frac{17}{110}\right)$$ $$e\left(\frac{39}{55}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{41}{110}\right)$$ $$e\left(\frac{43}{110}\right)$$ $$e\left(\frac{52}{55}\right)$$ $$e\left(\frac{32}{55}\right)$$
$$\chi_{1089}(37,\cdot)$$ 1089.v 55 no $$1$$ $$1$$ $$e\left(\frac{21}{55}\right)$$ $$e\left(\frac{42}{55}\right)$$ $$e\left(\frac{14}{55}\right)$$ $$e\left(\frac{37}{55}\right)$$ $$e\left(\frac{8}{55}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{31}{55}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{29}{55}\right)$$ $$e\left(\frac{39}{55}\right)$$
$$\chi_{1089}(38,\cdot)$$ 1089.bf 330 yes $$-1$$ $$1$$ $$e\left(\frac{307}{330}\right)$$ $$e\left(\frac{142}{165}\right)$$ $$e\left(\frac{113}{330}\right)$$ $$e\left(\frac{2}{165}\right)$$ $$e\left(\frac{87}{110}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{76}{165}\right)$$ $$e\left(\frac{311}{330}\right)$$ $$e\left(\frac{119}{165}\right)$$ $$e\left(\frac{101}{110}\right)$$
$$\chi_{1089}(40,\cdot)$$ 1089.s 30 no $$-1$$ $$1$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$-1$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{1089}(41,\cdot)$$ 1089.bd 330 yes $$1$$ $$1$$ $$e\left(\frac{7}{165}\right)$$ $$e\left(\frac{14}{165}\right)$$ $$e\left(\frac{211}{330}\right)$$ $$e\left(\frac{263}{330}\right)$$ $$e\left(\frac{7}{55}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{259}{330}\right)$$ $$e\left(\frac{277}{330}\right)$$ $$e\left(\frac{28}{165}\right)$$ $$e\left(\frac{41}{55}\right)$$
$$\chi_{1089}(43,\cdot)$$ 1089.x 66 yes $$-1$$ $$1$$ $$e\left(\frac{59}{66}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{17}{66}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{19}{66}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{3}{22}\right)$$
$$\chi_{1089}(46,\cdot)$$ 1089.ba 110 no $$-1$$ $$1$$ $$e\left(\frac{71}{110}\right)$$ $$e\left(\frac{16}{55}\right)$$ $$e\left(\frac{42}{55}\right)$$ $$e\left(\frac{57}{110}\right)$$ $$e\left(\frac{103}{110}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{21}{110}\right)$$ $$e\left(\frac{9}{55}\right)$$ $$e\left(\frac{32}{55}\right)$$ $$e\left(\frac{69}{110}\right)$$
$$\chi_{1089}(47,\cdot)$$ 1089.bf 330 yes $$-1$$ $$1$$ $$e\left(\frac{19}{330}\right)$$ $$e\left(\frac{19}{165}\right)$$ $$e\left(\frac{251}{330}\right)$$ $$e\left(\frac{149}{165}\right)$$ $$e\left(\frac{19}{110}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{52}{165}\right)$$ $$e\left(\frac{317}{330}\right)$$ $$e\left(\frac{38}{165}\right)$$ $$e\left(\frac{17}{110}\right)$$
$$\chi_{1089}(49,\cdot)$$ 1089.bc 165 yes $$1$$ $$1$$ $$e\left(\frac{76}{165}\right)$$ $$e\left(\frac{152}{165}\right)$$ $$e\left(\frac{14}{165}\right)$$ $$e\left(\frac{37}{165}\right)$$ $$e\left(\frac{21}{55}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{86}{165}\right)$$ $$e\left(\frac{113}{165}\right)$$ $$e\left(\frac{139}{165}\right)$$ $$e\left(\frac{13}{55}\right)$$
$$\chi_{1089}(50,\cdot)$$ 1089.bd 330 yes $$1$$ $$1$$ $$e\left(\frac{31}{165}\right)$$ $$e\left(\frac{62}{165}\right)$$ $$e\left(\frac{133}{330}\right)$$ $$e\left(\frac{269}{330}\right)$$ $$e\left(\frac{31}{55}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{157}{330}\right)$$ $$e\left(\frac{1}{330}\right)$$ $$e\left(\frac{124}{165}\right)$$ $$e\left(\frac{48}{55}\right)$$
$$\chi_{1089}(52,\cdot)$$ 1089.be 330 yes $$-1$$ $$1$$ $$e\left(\frac{199}{330}\right)$$ $$e\left(\frac{34}{165}\right)$$ $$e\left(\frac{103}{165}\right)$$ $$e\left(\frac{73}{330}\right)$$ $$e\left(\frac{89}{110}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{299}{330}\right)$$ $$e\left(\frac{136}{165}\right)$$ $$e\left(\frac{68}{165}\right)$$ $$e\left(\frac{97}{110}\right)$$