# Properties

 Modulus $1215$ Structure $$C_{2}\times C_{324}$$ Order $648$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(1215)

pari: g = idealstar(,1215,2)

## Character group

 sage: G.order()  pari: g.no Order = 648 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{324}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{1215}(731,\cdot)$, $\chi_{1215}(487,\cdot)$

## First 32 of 648 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$7$$ $$8$$ $$11$$ $$13$$ $$14$$ $$16$$ $$17$$ $$19$$
$$\chi_{1215}(1,\cdot)$$ 1215.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{1215}(2,\cdot)$$ 1215.bd 324 yes $$1$$ $$1$$ $$e\left(\frac{83}{324}\right)$$ $$e\left(\frac{83}{162}\right)$$ $$e\left(\frac{221}{324}\right)$$ $$e\left(\frac{83}{108}\right)$$ $$e\left(\frac{121}{162}\right)$$ $$e\left(\frac{259}{324}\right)$$ $$e\left(\frac{76}{81}\right)$$ $$e\left(\frac{2}{81}\right)$$ $$e\left(\frac{49}{108}\right)$$ $$e\left(\frac{25}{54}\right)$$
$$\chi_{1215}(4,\cdot)$$ 1215.ba 162 yes $$1$$ $$1$$ $$e\left(\frac{83}{162}\right)$$ $$e\left(\frac{2}{81}\right)$$ $$e\left(\frac{59}{162}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{40}{81}\right)$$ $$e\left(\frac{97}{162}\right)$$ $$e\left(\frac{71}{81}\right)$$ $$e\left(\frac{4}{81}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{25}{27}\right)$$
$$\chi_{1215}(7,\cdot)$$ 1215.bc 324 yes $$-1$$ $$1$$ $$e\left(\frac{221}{324}\right)$$ $$e\left(\frac{59}{162}\right)$$ $$e\left(\frac{161}{324}\right)$$ $$e\left(\frac{5}{108}\right)$$ $$e\left(\frac{23}{81}\right)$$ $$e\left(\frac{67}{324}\right)$$ $$e\left(\frac{29}{162}\right)$$ $$e\left(\frac{59}{81}\right)$$ $$e\left(\frac{55}{108}\right)$$ $$e\left(\frac{49}{54}\right)$$
$$\chi_{1215}(8,\cdot)$$ 1215.y 108 no $$1$$ $$1$$ $$e\left(\frac{83}{108}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{5}{108}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{43}{108}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$
$$\chi_{1215}(11,\cdot)$$ 1215.z 162 no $$-1$$ $$1$$ $$e\left(\frac{121}{162}\right)$$ $$e\left(\frac{40}{81}\right)$$ $$e\left(\frac{23}{81}\right)$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{61}{162}\right)$$ $$e\left(\frac{79}{81}\right)$$ $$e\left(\frac{5}{162}\right)$$ $$e\left(\frac{80}{81}\right)$$ $$e\left(\frac{35}{54}\right)$$ $$e\left(\frac{14}{27}\right)$$
$$\chi_{1215}(13,\cdot)$$ 1215.bc 324 yes $$-1$$ $$1$$ $$e\left(\frac{259}{324}\right)$$ $$e\left(\frac{97}{162}\right)$$ $$e\left(\frac{67}{324}\right)$$ $$e\left(\frac{43}{108}\right)$$ $$e\left(\frac{79}{81}\right)$$ $$e\left(\frac{209}{324}\right)$$ $$e\left(\frac{1}{162}\right)$$ $$e\left(\frac{16}{81}\right)$$ $$e\left(\frac{41}{108}\right)$$ $$e\left(\frac{11}{54}\right)$$
$$\chi_{1215}(14,\cdot)$$ 1215.bb 162 yes $$-1$$ $$1$$ $$e\left(\frac{76}{81}\right)$$ $$e\left(\frac{71}{81}\right)$$ $$e\left(\frac{29}{162}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{5}{162}\right)$$ $$e\left(\frac{1}{162}\right)$$ $$e\left(\frac{19}{162}\right)$$ $$e\left(\frac{61}{81}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$
$$\chi_{1215}(16,\cdot)$$ 1215.w 81 no $$1$$ $$1$$ $$e\left(\frac{2}{81}\right)$$ $$e\left(\frac{4}{81}\right)$$ $$e\left(\frac{59}{81}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{80}{81}\right)$$ $$e\left(\frac{16}{81}\right)$$ $$e\left(\frac{61}{81}\right)$$ $$e\left(\frac{8}{81}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$
$$\chi_{1215}(17,\cdot)$$ 1215.y 108 no $$1$$ $$1$$ $$e\left(\frac{49}{108}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{55}{108}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{35}{54}\right)$$ $$e\left(\frac{41}{108}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$
$$\chi_{1215}(19,\cdot)$$ 1215.t 54 no $$1$$ $$1$$ $$e\left(\frac{25}{54}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{11}{54}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{1215}(22,\cdot)$$ 1215.bc 324 yes $$-1$$ $$1$$ $$e\left(\frac{1}{324}\right)$$ $$e\left(\frac{1}{162}\right)$$ $$e\left(\frac{313}{324}\right)$$ $$e\left(\frac{1}{108}\right)$$ $$e\left(\frac{10}{81}\right)$$ $$e\left(\frac{251}{324}\right)$$ $$e\left(\frac{157}{162}\right)$$ $$e\left(\frac{1}{81}\right)$$ $$e\left(\frac{11}{108}\right)$$ $$e\left(\frac{53}{54}\right)$$
$$\chi_{1215}(23,\cdot)$$ 1215.bd 324 yes $$1$$ $$1$$ $$e\left(\frac{49}{324}\right)$$ $$e\left(\frac{49}{162}\right)$$ $$e\left(\frac{271}{324}\right)$$ $$e\left(\frac{49}{108}\right)$$ $$e\left(\frac{89}{162}\right)$$ $$e\left(\frac{149}{324}\right)$$ $$e\left(\frac{80}{81}\right)$$ $$e\left(\frac{49}{81}\right)$$ $$e\left(\frac{107}{108}\right)$$ $$e\left(\frac{5}{54}\right)$$
$$\chi_{1215}(26,\cdot)$$ 1215.o 18 no $$-1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{1215}(28,\cdot)$$ 1215.s 36 no $$-1$$ $$1$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{1215}(29,\cdot)$$ 1215.bb 162 yes $$-1$$ $$1$$ $$e\left(\frac{59}{81}\right)$$ $$e\left(\frac{37}{81}\right)$$ $$e\left(\frac{79}{162}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{103}{162}\right)$$ $$e\left(\frac{53}{162}\right)$$ $$e\left(\frac{35}{162}\right)$$ $$e\left(\frac{74}{81}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$
$$\chi_{1215}(31,\cdot)$$ 1215.w 81 no $$1$$ $$1$$ $$e\left(\frac{10}{81}\right)$$ $$e\left(\frac{20}{81}\right)$$ $$e\left(\frac{52}{81}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{76}{81}\right)$$ $$e\left(\frac{80}{81}\right)$$ $$e\left(\frac{62}{81}\right)$$ $$e\left(\frac{40}{81}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$
$$\chi_{1215}(32,\cdot)$$ 1215.bd 324 yes $$1$$ $$1$$ $$e\left(\frac{91}{324}\right)$$ $$e\left(\frac{91}{162}\right)$$ $$e\left(\frac{133}{324}\right)$$ $$e\left(\frac{91}{108}\right)$$ $$e\left(\frac{119}{162}\right)$$ $$e\left(\frac{323}{324}\right)$$ $$e\left(\frac{56}{81}\right)$$ $$e\left(\frac{10}{81}\right)$$ $$e\left(\frac{29}{108}\right)$$ $$e\left(\frac{17}{54}\right)$$
$$\chi_{1215}(34,\cdot)$$ 1215.ba 162 yes $$1$$ $$1$$ $$e\left(\frac{115}{162}\right)$$ $$e\left(\frac{34}{81}\right)$$ $$e\left(\frac{31}{162}\right)$$ $$e\left(\frac{7}{54}\right)$$ $$e\left(\frac{32}{81}\right)$$ $$e\left(\frac{29}{162}\right)$$ $$e\left(\frac{73}{81}\right)$$ $$e\left(\frac{68}{81}\right)$$ $$e\left(\frac{23}{54}\right)$$ $$e\left(\frac{20}{27}\right)$$
$$\chi_{1215}(37,\cdot)$$ 1215.x 108 no $$-1$$ $$1$$ $$e\left(\frac{91}{108}\right)$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{79}{108}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{53}{108}\right)$$ $$e\left(\frac{31}{54}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$
$$\chi_{1215}(38,\cdot)$$ 1215.bd 324 yes $$1$$ $$1$$ $$e\left(\frac{233}{324}\right)$$ $$e\left(\frac{71}{162}\right)$$ $$e\left(\frac{191}{324}\right)$$ $$e\left(\frac{17}{108}\right)$$ $$e\left(\frac{43}{162}\right)$$ $$e\left(\frac{1}{324}\right)$$ $$e\left(\frac{25}{81}\right)$$ $$e\left(\frac{71}{81}\right)$$ $$e\left(\frac{79}{108}\right)$$ $$e\left(\frac{37}{54}\right)$$
$$\chi_{1215}(41,\cdot)$$ 1215.z 162 no $$-1$$ $$1$$ $$e\left(\frac{53}{162}\right)$$ $$e\left(\frac{53}{81}\right)$$ $$e\left(\frac{73}{81}\right)$$ $$e\left(\frac{53}{54}\right)$$ $$e\left(\frac{95}{162}\right)$$ $$e\left(\frac{50}{81}\right)$$ $$e\left(\frac{37}{162}\right)$$ $$e\left(\frac{25}{81}\right)$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{1}{27}\right)$$
$$\chi_{1215}(43,\cdot)$$ 1215.bc 324 yes $$-1$$ $$1$$ $$e\left(\frac{179}{324}\right)$$ $$e\left(\frac{17}{162}\right)$$ $$e\left(\frac{299}{324}\right)$$ $$e\left(\frac{71}{108}\right)$$ $$e\left(\frac{8}{81}\right)$$ $$e\left(\frac{217}{324}\right)$$ $$e\left(\frac{77}{162}\right)$$ $$e\left(\frac{17}{81}\right)$$ $$e\left(\frac{25}{108}\right)$$ $$e\left(\frac{37}{54}\right)$$
$$\chi_{1215}(44,\cdot)$$ 1215.v 54 no $$-1$$ $$1$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{35}{54}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{47}{54}\right)$$ $$e\left(\frac{31}{54}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{1215}(46,\cdot)$$ 1215.q 27 no $$1$$ $$1$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{1215}(47,\cdot)$$ 1215.bd 324 yes $$1$$ $$1$$ $$e\left(\frac{203}{324}\right)$$ $$e\left(\frac{41}{162}\right)$$ $$e\left(\frac{197}{324}\right)$$ $$e\left(\frac{95}{108}\right)$$ $$e\left(\frac{91}{162}\right)$$ $$e\left(\frac{247}{324}\right)$$ $$e\left(\frac{19}{81}\right)$$ $$e\left(\frac{41}{81}\right)$$ $$e\left(\frac{73}{108}\right)$$ $$e\left(\frac{13}{54}\right)$$
$$\chi_{1215}(49,\cdot)$$ 1215.ba 162 yes $$1$$ $$1$$ $$e\left(\frac{59}{162}\right)$$ $$e\left(\frac{59}{81}\right)$$ $$e\left(\frac{161}{162}\right)$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{46}{81}\right)$$ $$e\left(\frac{67}{162}\right)$$ $$e\left(\frac{29}{81}\right)$$ $$e\left(\frac{37}{81}\right)$$ $$e\left(\frac{1}{54}\right)$$ $$e\left(\frac{22}{27}\right)$$
$$\chi_{1215}(52,\cdot)$$ 1215.bc 324 yes $$-1$$ $$1$$ $$e\left(\frac{101}{324}\right)$$ $$e\left(\frac{101}{162}\right)$$ $$e\left(\frac{185}{324}\right)$$ $$e\left(\frac{101}{108}\right)$$ $$e\left(\frac{38}{81}\right)$$ $$e\left(\frac{79}{324}\right)$$ $$e\left(\frac{143}{162}\right)$$ $$e\left(\frac{20}{81}\right)$$ $$e\left(\frac{31}{108}\right)$$ $$e\left(\frac{7}{54}\right)$$
$$\chi_{1215}(53,\cdot)$$ 1215.r 36 no $$1$$ $$1$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{1215}(56,\cdot)$$ 1215.z 162 no $$-1$$ $$1$$ $$e\left(\frac{73}{162}\right)$$ $$e\left(\frac{73}{81}\right)$$ $$e\left(\frac{44}{81}\right)$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{85}{162}\right)$$ $$e\left(\frac{49}{81}\right)$$ $$e\left(\frac{161}{162}\right)$$ $$e\left(\frac{65}{81}\right)$$ $$e\left(\frac{47}{54}\right)$$ $$e\left(\frac{8}{27}\right)$$
$$\chi_{1215}(58,\cdot)$$ 1215.bc 324 yes $$-1$$ $$1$$ $$e\left(\frac{319}{324}\right)$$ $$e\left(\frac{157}{162}\right)$$ $$e\left(\frac{55}{324}\right)$$ $$e\left(\frac{103}{108}\right)$$ $$e\left(\frac{31}{81}\right)$$ $$e\left(\frac{41}{324}\right)$$ $$e\left(\frac{25}{162}\right)$$ $$e\left(\frac{76}{81}\right)$$ $$e\left(\frac{53}{108}\right)$$ $$e\left(\frac{5}{54}\right)$$
$$\chi_{1215}(59,\cdot)$$ 1215.bb 162 yes $$-1$$ $$1$$ $$e\left(\frac{34}{81}\right)$$ $$e\left(\frac{68}{81}\right)$$ $$e\left(\frac{143}{162}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{47}{162}\right)$$ $$e\left(\frac{139}{162}\right)$$ $$e\left(\frac{49}{162}\right)$$ $$e\left(\frac{55}{81}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$