# Properties

 Modulus $1113$ Structure $$C_{156}\times C_{2}\times C_{2}$$ Order $624$

Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(1113)

pari: g = idealstar(,1113,2)

## Character group

 sage: G.order()  pari: g.no Order = 624 sage: H.invariants()  pari: g.cyc Structure = $$C_{156}\times C_{2}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{1113}(743,\cdot)$, $\chi_{1113}(955,\cdot)$, $\chi_{1113}(1009,\cdot)$

## First 32 of 624 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$
$$\chi_{1113}(1,\cdot)$$ 1113.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{1113}(2,\cdot)$$ 1113.bv 156 yes $$1$$ $$1$$ $$e\left(\frac{29}{156}\right)$$ $$e\left(\frac{29}{78}\right)$$ $$e\left(\frac{11}{156}\right)$$ $$e\left(\frac{29}{52}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{37}{39}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{59}{156}\right)$$
$$\chi_{1113}(4,\cdot)$$ 1113.bm 78 no $$1$$ $$1$$ $$e\left(\frac{29}{78}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{11}{78}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{20}{39}\right)$$ $$e\left(\frac{35}{39}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{19}{39}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{59}{78}\right)$$
$$\chi_{1113}(5,\cdot)$$ 1113.bt 156 yes $$-1$$ $$1$$ $$e\left(\frac{11}{156}\right)$$ $$e\left(\frac{11}{78}\right)$$ $$e\left(\frac{23}{156}\right)$$ $$e\left(\frac{11}{52}\right)$$ $$e\left(\frac{17}{78}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{11}{39}\right)$$ $$e\left(\frac{29}{78}\right)$$ $$e\left(\frac{95}{156}\right)$$
$$\chi_{1113}(8,\cdot)$$ 1113.bh 52 no $$1$$ $$1$$ $$e\left(\frac{29}{52}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{11}{52}\right)$$ $$e\left(\frac{35}{52}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{7}{52}\right)$$
$$\chi_{1113}(10,\cdot)$$ 1113.bq 78 no $$-1$$ $$1$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{20}{39}\right)$$ $$e\left(\frac{17}{78}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{37}{78}\right)$$ $$e\left(\frac{8}{39}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{31}{78}\right)$$ $$e\left(\frac{77}{78}\right)$$
$$\chi_{1113}(11,\cdot)$$ 1113.br 78 yes $$-1$$ $$1$$ $$e\left(\frac{37}{39}\right)$$ $$e\left(\frac{35}{39}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{8}{39}\right)$$ $$e\left(\frac{67}{78}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{31}{39}\right)$$ $$e\left(\frac{25}{78}\right)$$ $$e\left(\frac{47}{78}\right)$$
$$\chi_{1113}(13,\cdot)$$ 1113.ba 26 no $$-1$$ $$1$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{15}{26}\right)$$
$$\chi_{1113}(16,\cdot)$$ 1113.bg 39 no $$1$$ $$1$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{19}{39}\right)$$ $$e\left(\frac{11}{39}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{31}{39}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{38}{39}\right)$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{20}{39}\right)$$
$$\chi_{1113}(17,\cdot)$$ 1113.bp 78 yes $$1$$ $$1$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{29}{78}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{31}{78}\right)$$ $$e\left(\frac{25}{78}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{37}{39}\right)$$
$$\chi_{1113}(19,\cdot)$$ 1113.bs 156 no $$1$$ $$1$$ $$e\left(\frac{59}{156}\right)$$ $$e\left(\frac{59}{78}\right)$$ $$e\left(\frac{95}{156}\right)$$ $$e\left(\frac{7}{52}\right)$$ $$e\left(\frac{77}{78}\right)$$ $$e\left(\frac{47}{78}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{20}{39}\right)$$ $$e\left(\frac{37}{39}\right)$$ $$e\left(\frac{77}{156}\right)$$
$$\chi_{1113}(20,\cdot)$$ 1113.bj 52 yes $$-1$$ $$1$$ $$e\left(\frac{23}{52}\right)$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{15}{52}\right)$$ $$e\left(\frac{17}{52}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{19}{52}\right)$$
$$\chi_{1113}(22,\cdot)$$ 1113.bi 52 no $$-1$$ $$1$$ $$e\left(\frac{7}{52}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{17}{52}\right)$$ $$e\left(\frac{21}{52}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{51}{52}\right)$$
$$\chi_{1113}(23,\cdot)$$ 1113.u 12 yes $$1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$
$$\chi_{1113}(25,\cdot)$$ 1113.bm 78 no $$1$$ $$1$$ $$e\left(\frac{11}{78}\right)$$ $$e\left(\frac{11}{39}\right)$$ $$e\left(\frac{23}{78}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{20}{39}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{22}{39}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{17}{78}\right)$$
$$\chi_{1113}(26,\cdot)$$ 1113.bt 156 yes $$-1$$ $$1$$ $$e\left(\frac{101}{156}\right)$$ $$e\left(\frac{23}{78}\right)$$ $$e\left(\frac{41}{156}\right)$$ $$e\left(\frac{49}{52}\right)$$ $$e\left(\frac{71}{78}\right)$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{11}{78}\right)$$ $$e\left(\frac{149}{156}\right)$$
$$\chi_{1113}(29,\cdot)$$ 1113.z 26 no $$-1$$ $$1$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{19}{26}\right)$$
$$\chi_{1113}(31,\cdot)$$ 1113.bs 156 no $$1$$ $$1$$ $$e\left(\frac{151}{156}\right)$$ $$e\left(\frac{73}{78}\right)$$ $$e\left(\frac{103}{156}\right)$$ $$e\left(\frac{47}{52}\right)$$ $$e\left(\frac{49}{78}\right)$$ $$e\left(\frac{37}{78}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{34}{39}\right)$$ $$e\left(\frac{20}{39}\right)$$ $$e\left(\frac{49}{156}\right)$$
$$\chi_{1113}(32,\cdot)$$ 1113.bv 156 yes $$1$$ $$1$$ $$e\left(\frac{145}{156}\right)$$ $$e\left(\frac{67}{78}\right)$$ $$e\left(\frac{55}{156}\right)$$ $$e\left(\frac{41}{52}\right)$$ $$e\left(\frac{11}{39}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{139}{156}\right)$$
$$\chi_{1113}(34,\cdot)$$ 1113.bk 52 no $$1$$ $$1$$ $$e\left(\frac{11}{52}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{23}{52}\right)$$ $$e\left(\frac{33}{52}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{17}{52}\right)$$
$$\chi_{1113}(37,\cdot)$$ 1113.bm 78 no $$1$$ $$1$$ $$e\left(\frac{19}{78}\right)$$ $$e\left(\frac{19}{39}\right)$$ $$e\left(\frac{61}{78}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{31}{39}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{38}{39}\right)$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{1}{78}\right)$$
$$\chi_{1113}(38,\cdot)$$ 1113.bp 78 yes $$1$$ $$1$$ $$e\left(\frac{22}{39}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{53}{78}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{19}{78}\right)$$ $$e\left(\frac{43}{78}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{38}{39}\right)$$ $$e\left(\frac{34}{39}\right)$$
$$\chi_{1113}(40,\cdot)$$ 1113.bo 78 no $$-1$$ $$1$$ $$e\left(\frac{49}{78}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{77}{78}\right)$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{20}{39}\right)$$ $$e\left(\frac{35}{78}\right)$$ $$e\left(\frac{29}{39}\right)$$
$$\chi_{1113}(41,\cdot)$$ 1113.bj 52 yes $$-1$$ $$1$$ $$e\left(\frac{19}{52}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{35}{52}\right)$$ $$e\left(\frac{5}{52}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{27}{52}\right)$$
$$\chi_{1113}(43,\cdot)$$ 1113.be 26 no $$1$$ $$1$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{17}{26}\right)$$
$$\chi_{1113}(44,\cdot)$$ 1113.bl 78 yes $$-1$$ $$1$$ $$e\left(\frac{25}{78}\right)$$ $$e\left(\frac{25}{39}\right)$$ $$e\left(\frac{31}{78}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{59}{78}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{11}{39}\right)$$ $$e\left(\frac{29}{78}\right)$$ $$e\left(\frac{14}{39}\right)$$
$$\chi_{1113}(46,\cdot)$$ 1113.bg 39 no $$1$$ $$1$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{8}{39}\right)$$ $$e\left(\frac{19}{39}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{11}{39}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{16}{39}\right)$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{31}{39}\right)$$
$$\chi_{1113}(47,\cdot)$$ 1113.bn 78 yes $$1$$ $$1$$ $$e\left(\frac{1}{78}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{35}{78}\right)$$ $$e\left(\frac{71}{78}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{31}{39}\right)$$ $$e\left(\frac{37}{78}\right)$$
$$\chi_{1113}(50,\cdot)$$ 1113.bh 52 no $$1$$ $$1$$ $$e\left(\frac{17}{52}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{19}{52}\right)$$ $$e\left(\frac{51}{52}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{31}{52}\right)$$
$$\chi_{1113}(52,\cdot)$$ 1113.q 6 no $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{1113}(55,\cdot)$$ 1113.bk 52 no $$1$$ $$1$$ $$e\left(\frac{1}{52}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{21}{52}\right)$$ $$e\left(\frac{3}{52}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{11}{52}\right)$$
$$\chi_{1113}(58,\cdot)$$ 1113.bu 156 no $$-1$$ $$1$$ $$e\left(\frac{89}{156}\right)$$ $$e\left(\frac{11}{78}\right)$$ $$e\left(\frac{23}{156}\right)$$ $$e\left(\frac{37}{52}\right)$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{59}{78}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{11}{39}\right)$$ $$e\left(\frac{29}{78}\right)$$ $$e\left(\frac{17}{156}\right)$$