# Properties

 Modulus $729$ Structure $$C_{486}$$ Order $486$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(729)

pari: g = idealstar(,729,2)

## Character group

 sage: G.order()  pari: g.no Order = 486 sage: H.invariants()  pari: g.cyc Structure = $$C_{486}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{729}(2,\cdot)$

## First 32 of 486 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{729}(1,\cdot)$$ 729.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{729}(2,\cdot)$$ 729.l 486 yes $$-1$$ $$1$$ $$e\left(\frac{1}{486}\right)$$ $$e\left(\frac{1}{243}\right)$$ $$e\left(\frac{23}{486}\right)$$ $$e\left(\frac{197}{243}\right)$$ $$e\left(\frac{1}{162}\right)$$ $$e\left(\frac{4}{81}\right)$$ $$e\left(\frac{283}{486}\right)$$ $$e\left(\frac{166}{243}\right)$$ $$e\left(\frac{395}{486}\right)$$ $$e\left(\frac{2}{243}\right)$$
$$\chi_{729}(4,\cdot)$$ 729.k 243 yes $$1$$ $$1$$ $$e\left(\frac{1}{243}\right)$$ $$e\left(\frac{2}{243}\right)$$ $$e\left(\frac{23}{243}\right)$$ $$e\left(\frac{151}{243}\right)$$ $$e\left(\frac{1}{81}\right)$$ $$e\left(\frac{8}{81}\right)$$ $$e\left(\frac{40}{243}\right)$$ $$e\left(\frac{89}{243}\right)$$ $$e\left(\frac{152}{243}\right)$$ $$e\left(\frac{4}{243}\right)$$
$$\chi_{729}(5,\cdot)$$ 729.l 486 yes $$-1$$ $$1$$ $$e\left(\frac{23}{486}\right)$$ $$e\left(\frac{23}{243}\right)$$ $$e\left(\frac{43}{486}\right)$$ $$e\left(\frac{157}{243}\right)$$ $$e\left(\frac{23}{162}\right)$$ $$e\left(\frac{11}{81}\right)$$ $$e\left(\frac{191}{486}\right)$$ $$e\left(\frac{173}{243}\right)$$ $$e\left(\frac{337}{486}\right)$$ $$e\left(\frac{46}{243}\right)$$
$$\chi_{729}(7,\cdot)$$ 729.k 243 yes $$1$$ $$1$$ $$e\left(\frac{197}{243}\right)$$ $$e\left(\frac{151}{243}\right)$$ $$e\left(\frac{157}{243}\right)$$ $$e\left(\frac{101}{243}\right)$$ $$e\left(\frac{35}{81}\right)$$ $$e\left(\frac{37}{81}\right)$$ $$e\left(\frac{104}{243}\right)$$ $$e\left(\frac{37}{243}\right)$$ $$e\left(\frac{55}{243}\right)$$ $$e\left(\frac{59}{243}\right)$$
$$\chi_{729}(8,\cdot)$$ 729.j 162 no $$-1$$ $$1$$ $$e\left(\frac{1}{162}\right)$$ $$e\left(\frac{1}{81}\right)$$ $$e\left(\frac{23}{162}\right)$$ $$e\left(\frac{35}{81}\right)$$ $$e\left(\frac{1}{54}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{121}{162}\right)$$ $$e\left(\frac{4}{81}\right)$$ $$e\left(\frac{71}{162}\right)$$ $$e\left(\frac{2}{81}\right)$$
$$\chi_{729}(10,\cdot)$$ 729.i 81 no $$1$$ $$1$$ $$e\left(\frac{4}{81}\right)$$ $$e\left(\frac{8}{81}\right)$$ $$e\left(\frac{11}{81}\right)$$ $$e\left(\frac{37}{81}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{79}{81}\right)$$ $$e\left(\frac{32}{81}\right)$$ $$e\left(\frac{41}{81}\right)$$ $$e\left(\frac{16}{81}\right)$$
$$\chi_{729}(11,\cdot)$$ 729.l 486 yes $$-1$$ $$1$$ $$e\left(\frac{283}{486}\right)$$ $$e\left(\frac{40}{243}\right)$$ $$e\left(\frac{191}{486}\right)$$ $$e\left(\frac{104}{243}\right)$$ $$e\left(\frac{121}{162}\right)$$ $$e\left(\frac{79}{81}\right)$$ $$e\left(\frac{385}{486}\right)$$ $$e\left(\frac{79}{243}\right)$$ $$e\left(\frac{5}{486}\right)$$ $$e\left(\frac{80}{243}\right)$$
$$\chi_{729}(13,\cdot)$$ 729.k 243 yes $$1$$ $$1$$ $$e\left(\frac{166}{243}\right)$$ $$e\left(\frac{89}{243}\right)$$ $$e\left(\frac{173}{243}\right)$$ $$e\left(\frac{37}{243}\right)$$ $$e\left(\frac{4}{81}\right)$$ $$e\left(\frac{32}{81}\right)$$ $$e\left(\frac{79}{243}\right)$$ $$e\left(\frac{194}{243}\right)$$ $$e\left(\frac{203}{243}\right)$$ $$e\left(\frac{178}{243}\right)$$
$$\chi_{729}(14,\cdot)$$ 729.l 486 yes $$-1$$ $$1$$ $$e\left(\frac{395}{486}\right)$$ $$e\left(\frac{152}{243}\right)$$ $$e\left(\frac{337}{486}\right)$$ $$e\left(\frac{55}{243}\right)$$ $$e\left(\frac{71}{162}\right)$$ $$e\left(\frac{41}{81}\right)$$ $$e\left(\frac{5}{486}\right)$$ $$e\left(\frac{203}{243}\right)$$ $$e\left(\frac{19}{486}\right)$$ $$e\left(\frac{61}{243}\right)$$
$$\chi_{729}(16,\cdot)$$ 729.k 243 yes $$1$$ $$1$$ $$e\left(\frac{2}{243}\right)$$ $$e\left(\frac{4}{243}\right)$$ $$e\left(\frac{46}{243}\right)$$ $$e\left(\frac{59}{243}\right)$$ $$e\left(\frac{2}{81}\right)$$ $$e\left(\frac{16}{81}\right)$$ $$e\left(\frac{80}{243}\right)$$ $$e\left(\frac{178}{243}\right)$$ $$e\left(\frac{61}{243}\right)$$ $$e\left(\frac{8}{243}\right)$$
$$\chi_{729}(17,\cdot)$$ 729.j 162 no $$-1$$ $$1$$ $$e\left(\frac{11}{162}\right)$$ $$e\left(\frac{11}{81}\right)$$ $$e\left(\frac{91}{162}\right)$$ $$e\left(\frac{61}{81}\right)$$ $$e\left(\frac{11}{54}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{35}{162}\right)$$ $$e\left(\frac{44}{81}\right)$$ $$e\left(\frac{133}{162}\right)$$ $$e\left(\frac{22}{81}\right)$$
$$\chi_{729}(19,\cdot)$$ 729.i 81 no $$1$$ $$1$$ $$e\left(\frac{53}{81}\right)$$ $$e\left(\frac{25}{81}\right)$$ $$e\left(\frac{4}{81}\right)$$ $$e\left(\frac{65}{81}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{14}{81}\right)$$ $$e\left(\frac{19}{81}\right)$$ $$e\left(\frac{37}{81}\right)$$ $$e\left(\frac{50}{81}\right)$$
$$\chi_{729}(20,\cdot)$$ 729.l 486 yes $$-1$$ $$1$$ $$e\left(\frac{25}{486}\right)$$ $$e\left(\frac{25}{243}\right)$$ $$e\left(\frac{89}{486}\right)$$ $$e\left(\frac{65}{243}\right)$$ $$e\left(\frac{25}{162}\right)$$ $$e\left(\frac{19}{81}\right)$$ $$e\left(\frac{271}{486}\right)$$ $$e\left(\frac{19}{243}\right)$$ $$e\left(\frac{155}{486}\right)$$ $$e\left(\frac{50}{243}\right)$$
$$\chi_{729}(22,\cdot)$$ 729.k 243 yes $$1$$ $$1$$ $$e\left(\frac{142}{243}\right)$$ $$e\left(\frac{41}{243}\right)$$ $$e\left(\frac{107}{243}\right)$$ $$e\left(\frac{58}{243}\right)$$ $$e\left(\frac{61}{81}\right)$$ $$e\left(\frac{2}{81}\right)$$ $$e\left(\frac{91}{243}\right)$$ $$e\left(\frac{2}{243}\right)$$ $$e\left(\frac{200}{243}\right)$$ $$e\left(\frac{82}{243}\right)$$
$$\chi_{729}(23,\cdot)$$ 729.l 486 yes $$-1$$ $$1$$ $$e\left(\frac{389}{486}\right)$$ $$e\left(\frac{146}{243}\right)$$ $$e\left(\frac{199}{486}\right)$$ $$e\left(\frac{88}{243}\right)$$ $$e\left(\frac{65}{162}\right)$$ $$e\left(\frac{17}{81}\right)$$ $$e\left(\frac{251}{486}\right)$$ $$e\left(\frac{179}{243}\right)$$ $$e\left(\frac{79}{486}\right)$$ $$e\left(\frac{49}{243}\right)$$
$$\chi_{729}(25,\cdot)$$ 729.k 243 yes $$1$$ $$1$$ $$e\left(\frac{23}{243}\right)$$ $$e\left(\frac{46}{243}\right)$$ $$e\left(\frac{43}{243}\right)$$ $$e\left(\frac{71}{243}\right)$$ $$e\left(\frac{23}{81}\right)$$ $$e\left(\frac{22}{81}\right)$$ $$e\left(\frac{191}{243}\right)$$ $$e\left(\frac{103}{243}\right)$$ $$e\left(\frac{94}{243}\right)$$ $$e\left(\frac{92}{243}\right)$$
$$\chi_{729}(26,\cdot)$$ 729.h 54 no $$-1$$ $$1$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{35}{54}\right)$$ $$e\left(\frac{20}{27}\right)$$
$$\chi_{729}(28,\cdot)$$ 729.g 27 no $$1$$ $$1$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$
$$\chi_{729}(29,\cdot)$$ 729.l 486 yes $$-1$$ $$1$$ $$e\left(\frac{199}{486}\right)$$ $$e\left(\frac{199}{243}\right)$$ $$e\left(\frac{203}{486}\right)$$ $$e\left(\frac{80}{243}\right)$$ $$e\left(\frac{37}{162}\right)$$ $$e\left(\frac{67}{81}\right)$$ $$e\left(\frac{427}{486}\right)$$ $$e\left(\frac{229}{243}\right)$$ $$e\left(\frac{359}{486}\right)$$ $$e\left(\frac{155}{243}\right)$$
$$\chi_{729}(31,\cdot)$$ 729.k 243 yes $$1$$ $$1$$ $$e\left(\frac{172}{243}\right)$$ $$e\left(\frac{101}{243}\right)$$ $$e\left(\frac{68}{243}\right)$$ $$e\left(\frac{214}{243}\right)$$ $$e\left(\frac{10}{81}\right)$$ $$e\left(\frac{80}{81}\right)$$ $$e\left(\frac{76}{243}\right)$$ $$e\left(\frac{242}{243}\right)$$ $$e\left(\frac{143}{243}\right)$$ $$e\left(\frac{202}{243}\right)$$
$$\chi_{729}(32,\cdot)$$ 729.l 486 yes $$-1$$ $$1$$ $$e\left(\frac{5}{486}\right)$$ $$e\left(\frac{5}{243}\right)$$ $$e\left(\frac{115}{486}\right)$$ $$e\left(\frac{13}{243}\right)$$ $$e\left(\frac{5}{162}\right)$$ $$e\left(\frac{20}{81}\right)$$ $$e\left(\frac{443}{486}\right)$$ $$e\left(\frac{101}{243}\right)$$ $$e\left(\frac{31}{486}\right)$$ $$e\left(\frac{10}{243}\right)$$
$$\chi_{729}(34,\cdot)$$ 729.k 243 yes $$1$$ $$1$$ $$e\left(\frac{17}{243}\right)$$ $$e\left(\frac{34}{243}\right)$$ $$e\left(\frac{148}{243}\right)$$ $$e\left(\frac{137}{243}\right)$$ $$e\left(\frac{17}{81}\right)$$ $$e\left(\frac{55}{81}\right)$$ $$e\left(\frac{194}{243}\right)$$ $$e\left(\frac{55}{243}\right)$$ $$e\left(\frac{154}{243}\right)$$ $$e\left(\frac{68}{243}\right)$$
$$\chi_{729}(35,\cdot)$$ 729.j 162 no $$-1$$ $$1$$ $$e\left(\frac{139}{162}\right)$$ $$e\left(\frac{58}{81}\right)$$ $$e\left(\frac{119}{162}\right)$$ $$e\left(\frac{5}{81}\right)$$ $$e\left(\frac{31}{54}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{133}{162}\right)$$ $$e\left(\frac{70}{81}\right)$$ $$e\left(\frac{149}{162}\right)$$ $$e\left(\frac{35}{81}\right)$$
$$\chi_{729}(37,\cdot)$$ 729.i 81 no $$1$$ $$1$$ $$e\left(\frac{43}{81}\right)$$ $$e\left(\frac{5}{81}\right)$$ $$e\left(\frac{17}{81}\right)$$ $$e\left(\frac{13}{81}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{19}{81}\right)$$ $$e\left(\frac{20}{81}\right)$$ $$e\left(\frac{56}{81}\right)$$ $$e\left(\frac{10}{81}\right)$$
$$\chi_{729}(38,\cdot)$$ 729.l 486 yes $$-1$$ $$1$$ $$e\left(\frac{319}{486}\right)$$ $$e\left(\frac{76}{243}\right)$$ $$e\left(\frac{47}{486}\right)$$ $$e\left(\frac{149}{243}\right)$$ $$e\left(\frac{157}{162}\right)$$ $$e\left(\frac{61}{81}\right)$$ $$e\left(\frac{367}{486}\right)$$ $$e\left(\frac{223}{243}\right)$$ $$e\left(\frac{131}{486}\right)$$ $$e\left(\frac{152}{243}\right)$$
$$\chi_{729}(40,\cdot)$$ 729.k 243 yes $$1$$ $$1$$ $$e\left(\frac{13}{243}\right)$$ $$e\left(\frac{26}{243}\right)$$ $$e\left(\frac{56}{243}\right)$$ $$e\left(\frac{19}{243}\right)$$ $$e\left(\frac{13}{81}\right)$$ $$e\left(\frac{23}{81}\right)$$ $$e\left(\frac{34}{243}\right)$$ $$e\left(\frac{185}{243}\right)$$ $$e\left(\frac{32}{243}\right)$$ $$e\left(\frac{52}{243}\right)$$
$$\chi_{729}(41,\cdot)$$ 729.l 486 yes $$-1$$ $$1$$ $$e\left(\frac{215}{486}\right)$$ $$e\left(\frac{215}{243}\right)$$ $$e\left(\frac{85}{486}\right)$$ $$e\left(\frac{73}{243}\right)$$ $$e\left(\frac{53}{162}\right)$$ $$e\left(\frac{50}{81}\right)$$ $$e\left(\frac{95}{486}\right)$$ $$e\left(\frac{212}{243}\right)$$ $$e\left(\frac{361}{486}\right)$$ $$e\left(\frac{187}{243}\right)$$
$$\chi_{729}(43,\cdot)$$ 729.k 243 yes $$1$$ $$1$$ $$e\left(\frac{227}{243}\right)$$ $$e\left(\frac{211}{243}\right)$$ $$e\left(\frac{118}{243}\right)$$ $$e\left(\frac{14}{243}\right)$$ $$e\left(\frac{65}{81}\right)$$ $$e\left(\frac{34}{81}\right)$$ $$e\left(\frac{89}{243}\right)$$ $$e\left(\frac{34}{243}\right)$$ $$e\left(\frac{241}{243}\right)$$ $$e\left(\frac{179}{243}\right)$$
$$\chi_{729}(44,\cdot)$$ 729.j 162 no $$-1$$ $$1$$ $$e\left(\frac{95}{162}\right)$$ $$e\left(\frac{14}{81}\right)$$ $$e\left(\frac{79}{162}\right)$$ $$e\left(\frac{4}{81}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{155}{162}\right)$$ $$e\left(\frac{56}{81}\right)$$ $$e\left(\frac{103}{162}\right)$$ $$e\left(\frac{28}{81}\right)$$
$$\chi_{729}(46,\cdot)$$ 729.i 81 no $$1$$ $$1$$ $$e\left(\frac{65}{81}\right)$$ $$e\left(\frac{49}{81}\right)$$ $$e\left(\frac{37}{81}\right)$$ $$e\left(\frac{14}{81}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{8}{81}\right)$$ $$e\left(\frac{34}{81}\right)$$ $$e\left(\frac{79}{81}\right)$$ $$e\left(\frac{17}{81}\right)$$
$$\chi_{729}(47,\cdot)$$ 729.l 486 yes $$-1$$ $$1$$ $$e\left(\frac{385}{486}\right)$$ $$e\left(\frac{142}{243}\right)$$ $$e\left(\frac{107}{486}\right)$$ $$e\left(\frac{29}{243}\right)$$ $$e\left(\frac{61}{162}\right)$$ $$e\left(\frac{1}{81}\right)$$ $$e\left(\frac{91}{486}\right)$$ $$e\left(\frac{1}{243}\right)$$ $$e\left(\frac{443}{486}\right)$$ $$e\left(\frac{41}{243}\right)$$
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