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 | \(-1\) | \(1\) | \(e\left(\frac{385}{486}\right)\) | \(e\left(\frac{142}{243}\right)\) | \(e\left(\frac{107}{486}\right)\) | \(e\left(\frac{29}{243}\right)\) | \(e\left(\frac{61}{162}\right)\) | \(e\left(\frac{1}{81}\right)\) | \(e\left(\frac{91}{486}\right)\) | \(e\left(\frac{1}{243}\right)\) | \(e\left(\frac{443}{486}\right)\) | \(e\left(\frac{41}{243}\right)\) |