# Properties

 Modulus $869$ Structure $$C_{390}\times C_{2}$$ Order $780$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(869)

pari: g = idealstar(,869,2)

## Character group

 sage: G.order()  pari: g.no Order = 780 sage: H.invariants()  pari: g.cyc Structure = $$C_{390}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{869}(475,\cdot)$, $\chi_{869}(793,\cdot)$

## First 32 of 780 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$12$$
$$\chi_{869}(1,\cdot)$$ 869.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{869}(2,\cdot)$$ 869.be 390 yes $$-1$$ $$1$$ $$e\left(\frac{119}{390}\right)$$ $$e\left(\frac{166}{195}\right)$$ $$e\left(\frac{119}{195}\right)$$ $$e\left(\frac{113}{195}\right)$$ $$e\left(\frac{61}{390}\right)$$ $$e\left(\frac{163}{390}\right)$$ $$e\left(\frac{119}{130}\right)$$ $$e\left(\frac{137}{195}\right)$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{6}{13}\right)$$
$$\chi_{869}(3,\cdot)$$ 869.bd 390 yes $$-1$$ $$1$$ $$e\left(\frac{166}{195}\right)$$ $$e\left(\frac{161}{390}\right)$$ $$e\left(\frac{137}{195}\right)$$ $$e\left(\frac{194}{195}\right)$$ $$e\left(\frac{103}{390}\right)$$ $$e\left(\frac{109}{390}\right)$$ $$e\left(\frac{36}{65}\right)$$ $$e\left(\frac{161}{195}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{3}{26}\right)$$
$$\chi_{869}(4,\cdot)$$ 869.bc 195 yes $$1$$ $$1$$ $$e\left(\frac{119}{195}\right)$$ $$e\left(\frac{137}{195}\right)$$ $$e\left(\frac{43}{195}\right)$$ $$e\left(\frac{31}{195}\right)$$ $$e\left(\frac{61}{195}\right)$$ $$e\left(\frac{163}{195}\right)$$ $$e\left(\frac{54}{65}\right)$$ $$e\left(\frac{79}{195}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$
$$\chi_{869}(5,\cdot)$$ 869.bc 195 yes $$1$$ $$1$$ $$e\left(\frac{113}{195}\right)$$ $$e\left(\frac{194}{195}\right)$$ $$e\left(\frac{31}{195}\right)$$ $$e\left(\frac{172}{195}\right)$$ $$e\left(\frac{112}{195}\right)$$ $$e\left(\frac{181}{195}\right)$$ $$e\left(\frac{48}{65}\right)$$ $$e\left(\frac{193}{195}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$
$$\chi_{869}(6,\cdot)$$ 869.bf 390 yes $$1$$ $$1$$ $$e\left(\frac{61}{390}\right)$$ $$e\left(\frac{103}{390}\right)$$ $$e\left(\frac{61}{195}\right)$$ $$e\left(\frac{112}{195}\right)$$ $$e\left(\frac{82}{195}\right)$$ $$e\left(\frac{136}{195}\right)$$ $$e\left(\frac{61}{130}\right)$$ $$e\left(\frac{103}{195}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{15}{26}\right)$$
$$\chi_{869}(7,\cdot)$$ 869.bf 390 yes $$1$$ $$1$$ $$e\left(\frac{163}{390}\right)$$ $$e\left(\frac{109}{390}\right)$$ $$e\left(\frac{163}{195}\right)$$ $$e\left(\frac{181}{195}\right)$$ $$e\left(\frac{136}{195}\right)$$ $$e\left(\frac{178}{195}\right)$$ $$e\left(\frac{33}{130}\right)$$ $$e\left(\frac{109}{195}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{3}{26}\right)$$
$$\chi_{869}(8,\cdot)$$ 869.ba 130 yes $$-1$$ $$1$$ $$e\left(\frac{119}{130}\right)$$ $$e\left(\frac{36}{65}\right)$$ $$e\left(\frac{54}{65}\right)$$ $$e\left(\frac{48}{65}\right)$$ $$e\left(\frac{61}{130}\right)$$ $$e\left(\frac{33}{130}\right)$$ $$e\left(\frac{97}{130}\right)$$ $$e\left(\frac{7}{65}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{5}{13}\right)$$
$$\chi_{869}(9,\cdot)$$ 869.bc 195 yes $$1$$ $$1$$ $$e\left(\frac{137}{195}\right)$$ $$e\left(\frac{161}{195}\right)$$ $$e\left(\frac{79}{195}\right)$$ $$e\left(\frac{193}{195}\right)$$ $$e\left(\frac{103}{195}\right)$$ $$e\left(\frac{109}{195}\right)$$ $$e\left(\frac{7}{65}\right)$$ $$e\left(\frac{127}{195}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$
$$\chi_{869}(10,\cdot)$$ 869.p 26 yes $$-1$$ $$1$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{8}{13}\right)$$
$$\chi_{869}(12,\cdot)$$ 869.o 26 no $$-1$$ $$1$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{1}{26}\right)$$
$$\chi_{869}(13,\cdot)$$ 869.be 390 yes $$-1$$ $$1$$ $$e\left(\frac{329}{390}\right)$$ $$e\left(\frac{46}{195}\right)$$ $$e\left(\frac{134}{195}\right)$$ $$e\left(\frac{83}{195}\right)$$ $$e\left(\frac{31}{390}\right)$$ $$e\left(\frac{313}{390}\right)$$ $$e\left(\frac{69}{130}\right)$$ $$e\left(\frac{92}{195}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{12}{13}\right)$$
$$\chi_{869}(14,\cdot)$$ 869.bb 130 yes $$-1$$ $$1$$ $$e\left(\frac{47}{65}\right)$$ $$e\left(\frac{17}{130}\right)$$ $$e\left(\frac{29}{65}\right)$$ $$e\left(\frac{33}{65}\right)$$ $$e\left(\frac{111}{130}\right)$$ $$e\left(\frac{43}{130}\right)$$ $$e\left(\frac{11}{65}\right)$$ $$e\left(\frac{17}{65}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{15}{26}\right)$$
$$\chi_{869}(15,\cdot)$$ 869.bb 130 yes $$-1$$ $$1$$ $$e\left(\frac{28}{65}\right)$$ $$e\left(\frac{53}{130}\right)$$ $$e\left(\frac{56}{65}\right)$$ $$e\left(\frac{57}{65}\right)$$ $$e\left(\frac{109}{130}\right)$$ $$e\left(\frac{27}{130}\right)$$ $$e\left(\frac{19}{65}\right)$$ $$e\left(\frac{53}{65}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{7}{26}\right)$$
$$\chi_{869}(16,\cdot)$$ 869.bc 195 yes $$1$$ $$1$$ $$e\left(\frac{43}{195}\right)$$ $$e\left(\frac{79}{195}\right)$$ $$e\left(\frac{86}{195}\right)$$ $$e\left(\frac{62}{195}\right)$$ $$e\left(\frac{122}{195}\right)$$ $$e\left(\frac{131}{195}\right)$$ $$e\left(\frac{43}{65}\right)$$ $$e\left(\frac{158}{195}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$
$$\chi_{869}(17,\cdot)$$ 869.z 130 yes $$1$$ $$1$$ $$e\left(\frac{127}{130}\right)$$ $$e\left(\frac{61}{130}\right)$$ $$e\left(\frac{62}{65}\right)$$ $$e\left(\frac{19}{65}\right)$$ $$e\left(\frac{29}{65}\right)$$ $$e\left(\frac{37}{65}\right)$$ $$e\left(\frac{121}{130}\right)$$ $$e\left(\frac{61}{65}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{11}{26}\right)$$
$$\chi_{869}(18,\cdot)$$ 869.ba 130 yes $$-1$$ $$1$$ $$e\left(\frac{1}{130}\right)$$ $$e\left(\frac{44}{65}\right)$$ $$e\left(\frac{1}{65}\right)$$ $$e\left(\frac{37}{65}\right)$$ $$e\left(\frac{89}{130}\right)$$ $$e\left(\frac{127}{130}\right)$$ $$e\left(\frac{3}{130}\right)$$ $$e\left(\frac{23}{65}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{9}{13}\right)$$
$$\chi_{869}(19,\cdot)$$ 869.be 390 yes $$-1$$ $$1$$ $$e\left(\frac{367}{390}\right)$$ $$e\left(\frac{158}{195}\right)$$ $$e\left(\frac{172}{195}\right)$$ $$e\left(\frac{124}{195}\right)$$ $$e\left(\frac{293}{390}\right)$$ $$e\left(\frac{329}{390}\right)$$ $$e\left(\frac{107}{130}\right)$$ $$e\left(\frac{121}{195}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{9}{13}\right)$$
$$\chi_{869}(20,\cdot)$$ 869.bc 195 yes $$1$$ $$1$$ $$e\left(\frac{37}{195}\right)$$ $$e\left(\frac{136}{195}\right)$$ $$e\left(\frac{74}{195}\right)$$ $$e\left(\frac{8}{195}\right)$$ $$e\left(\frac{173}{195}\right)$$ $$e\left(\frac{149}{195}\right)$$ $$e\left(\frac{37}{65}\right)$$ $$e\left(\frac{77}{195}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$
$$\chi_{869}(21,\cdot)$$ 869.p 26 yes $$-1$$ $$1$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{3}{13}\right)$$
$$\chi_{869}(23,\cdot)$$ 869.e 3 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$1$$
$$\chi_{869}(24,\cdot)$$ 869.r 30 yes $$1$$ $$1$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$-1$$ $$-1$$
$$\chi_{869}(25,\cdot)$$ 869.bc 195 yes $$1$$ $$1$$ $$e\left(\frac{31}{195}\right)$$ $$e\left(\frac{193}{195}\right)$$ $$e\left(\frac{62}{195}\right)$$ $$e\left(\frac{149}{195}\right)$$ $$e\left(\frac{29}{195}\right)$$ $$e\left(\frac{167}{195}\right)$$ $$e\left(\frac{31}{65}\right)$$ $$e\left(\frac{191}{195}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$
$$\chi_{869}(26,\cdot)$$ 869.bc 195 yes $$1$$ $$1$$ $$e\left(\frac{29}{195}\right)$$ $$e\left(\frac{17}{195}\right)$$ $$e\left(\frac{58}{195}\right)$$ $$e\left(\frac{1}{195}\right)$$ $$e\left(\frac{46}{195}\right)$$ $$e\left(\frac{43}{195}\right)$$ $$e\left(\frac{29}{65}\right)$$ $$e\left(\frac{34}{195}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{5}{13}\right)$$
$$\chi_{869}(27,\cdot)$$ 869.bb 130 yes $$-1$$ $$1$$ $$e\left(\frac{36}{65}\right)$$ $$e\left(\frac{31}{130}\right)$$ $$e\left(\frac{7}{65}\right)$$ $$e\left(\frac{64}{65}\right)$$ $$e\left(\frac{103}{130}\right)$$ $$e\left(\frac{109}{130}\right)$$ $$e\left(\frac{43}{65}\right)$$ $$e\left(\frac{31}{65}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{9}{26}\right)$$
$$\chi_{869}(28,\cdot)$$ 869.bf 390 yes $$1$$ $$1$$ $$e\left(\frac{11}{390}\right)$$ $$e\left(\frac{383}{390}\right)$$ $$e\left(\frac{11}{195}\right)$$ $$e\left(\frac{17}{195}\right)$$ $$e\left(\frac{2}{195}\right)$$ $$e\left(\frac{146}{195}\right)$$ $$e\left(\frac{11}{130}\right)$$ $$e\left(\frac{188}{195}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{1}{26}\right)$$
$$\chi_{869}(29,\cdot)$$ 869.bf 390 yes $$1$$ $$1$$ $$e\left(\frac{103}{390}\right)$$ $$e\left(\frac{289}{390}\right)$$ $$e\left(\frac{103}{195}\right)$$ $$e\left(\frac{106}{195}\right)$$ $$e\left(\frac{1}{195}\right)$$ $$e\left(\frac{73}{195}\right)$$ $$e\left(\frac{103}{130}\right)$$ $$e\left(\frac{94}{195}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{7}{26}\right)$$
$$\chi_{869}(30,\cdot)$$ 869.bf 390 yes $$1$$ $$1$$ $$e\left(\frac{287}{390}\right)$$ $$e\left(\frac{101}{390}\right)$$ $$e\left(\frac{92}{195}\right)$$ $$e\left(\frac{89}{195}\right)$$ $$e\left(\frac{194}{195}\right)$$ $$e\left(\frac{122}{195}\right)$$ $$e\left(\frac{27}{130}\right)$$ $$e\left(\frac{101}{195}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{19}{26}\right)$$
$$\chi_{869}(31,\cdot)$$ 869.bc 195 yes $$1$$ $$1$$ $$e\left(\frac{92}{195}\right)$$ $$e\left(\frac{101}{195}\right)$$ $$e\left(\frac{184}{195}\right)$$ $$e\left(\frac{178}{195}\right)$$ $$e\left(\frac{193}{195}\right)$$ $$e\left(\frac{49}{195}\right)$$ $$e\left(\frac{27}{65}\right)$$ $$e\left(\frac{7}{195}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$
$$\chi_{869}(32,\cdot)$$ 869.x 78 yes $$-1$$ $$1$$ $$e\left(\frac{41}{78}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{35}{39}\right)$$ $$e\left(\frac{61}{78}\right)$$ $$e\left(\frac{7}{78}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{20}{39}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{4}{13}\right)$$
$$\chi_{869}(34,\cdot)$$ 869.y 78 no $$-1$$ $$1$$ $$e\left(\frac{11}{39}\right)$$ $$e\left(\frac{25}{78}\right)$$ $$e\left(\frac{22}{39}\right)$$ $$e\left(\frac{34}{39}\right)$$ $$e\left(\frac{47}{78}\right)$$ $$e\left(\frac{77}{78}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{25}{39}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{23}{26}\right)$$
$$\chi_{869}(35,\cdot)$$ 869.bf 390 yes $$1$$ $$1$$ $$e\left(\frac{389}{390}\right)$$ $$e\left(\frac{107}{390}\right)$$ $$e\left(\frac{194}{195}\right)$$ $$e\left(\frac{158}{195}\right)$$ $$e\left(\frac{53}{195}\right)$$ $$e\left(\frac{164}{195}\right)$$ $$e\left(\frac{129}{130}\right)$$ $$e\left(\frac{107}{195}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{7}{26}\right)$$