from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(643, base_ring=CyclotomicField(642))
M = H._module
chi = DirichletCharacter(H, M([530]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,643))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(643\) | |
| Conductor: | \(643\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(321\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{321})$ |
| Fixed field: | Number field defined by a degree 321 polynomial (not computed) |
First 31 of 212 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{643}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{104}{107}\right)\) | \(e\left(\frac{10}{107}\right)\) | \(e\left(\frac{101}{107}\right)\) | \(e\left(\frac{31}{107}\right)\) | \(e\left(\frac{7}{107}\right)\) | \(e\left(\frac{173}{321}\right)\) | \(e\left(\frac{98}{107}\right)\) | \(e\left(\frac{20}{107}\right)\) | \(e\left(\frac{28}{107}\right)\) | \(e\left(\frac{265}{321}\right)\) |
| \(\chi_{643}(22,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{107}\right)\) | \(e\left(\frac{55}{107}\right)\) | \(e\left(\frac{74}{107}\right)\) | \(e\left(\frac{10}{107}\right)\) | \(e\left(\frac{92}{107}\right)\) | \(e\left(\frac{256}{321}\right)\) | \(e\left(\frac{4}{107}\right)\) | \(e\left(\frac{3}{107}\right)\) | \(e\left(\frac{47}{107}\right)\) | \(e\left(\frac{227}{321}\right)\) |
| \(\chi_{643}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{80}{107}\right)\) | \(e\left(\frac{90}{107}\right)\) | \(e\left(\frac{53}{107}\right)\) | \(e\left(\frac{65}{107}\right)\) | \(e\left(\frac{63}{107}\right)\) | \(e\left(\frac{166}{321}\right)\) | \(e\left(\frac{26}{107}\right)\) | \(e\left(\frac{73}{107}\right)\) | \(e\left(\frac{38}{107}\right)\) | \(e\left(\frac{245}{321}\right)\) |
| \(\chi_{643}(26,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{107}\right)\) | \(e\left(\frac{11}{107}\right)\) | \(e\left(\frac{79}{107}\right)\) | \(e\left(\frac{2}{107}\right)\) | \(e\left(\frac{104}{107}\right)\) | \(e\left(\frac{308}{321}\right)\) | \(e\left(\frac{65}{107}\right)\) | \(e\left(\frac{22}{107}\right)\) | \(e\left(\frac{95}{107}\right)\) | \(e\left(\frac{238}{321}\right)\) |
| \(\chi_{643}(28,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{107}\right)\) | \(e\left(\frac{17}{107}\right)\) | \(e\left(\frac{54}{107}\right)\) | \(e\left(\frac{42}{107}\right)\) | \(e\left(\frac{44}{107}\right)\) | \(e\left(\frac{155}{321}\right)\) | \(e\left(\frac{81}{107}\right)\) | \(e\left(\frac{34}{107}\right)\) | \(e\left(\frac{69}{107}\right)\) | \(e\left(\frac{76}{321}\right)\) |
| \(\chi_{643}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{107}\right)\) | \(e\left(\frac{105}{107}\right)\) | \(e\left(\frac{44}{107}\right)\) | \(e\left(\frac{58}{107}\right)\) | \(e\left(\frac{20}{107}\right)\) | \(e\left(\frac{158}{321}\right)\) | \(e\left(\frac{66}{107}\right)\) | \(e\left(\frac{103}{107}\right)\) | \(e\left(\frac{80}{107}\right)\) | \(e\left(\frac{268}{321}\right)\) |
| \(\chi_{643}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{98}{107}\right)\) | \(e\left(\frac{30}{107}\right)\) | \(e\left(\frac{89}{107}\right)\) | \(e\left(\frac{93}{107}\right)\) | \(e\left(\frac{21}{107}\right)\) | \(e\left(\frac{91}{321}\right)\) | \(e\left(\frac{80}{107}\right)\) | \(e\left(\frac{60}{107}\right)\) | \(e\left(\frac{84}{107}\right)\) | \(e\left(\frac{260}{321}\right)\) |
| \(\chi_{643}(33,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{107}\right)\) | \(e\left(\frac{22}{107}\right)\) | \(e\left(\frac{51}{107}\right)\) | \(e\left(\frac{4}{107}\right)\) | \(e\left(\frac{101}{107}\right)\) | \(e\left(\frac{295}{321}\right)\) | \(e\left(\frac{23}{107}\right)\) | \(e\left(\frac{44}{107}\right)\) | \(e\left(\frac{83}{107}\right)\) | \(e\left(\frac{155}{321}\right)\) |
| \(\chi_{643}(34,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{76}{107}\right)\) | \(e\left(\frac{32}{107}\right)\) | \(e\left(\frac{45}{107}\right)\) | \(e\left(\frac{35}{107}\right)\) | \(e\left(\frac{1}{107}\right)\) | \(e\left(\frac{40}{321}\right)\) | \(e\left(\frac{14}{107}\right)\) | \(e\left(\frac{64}{107}\right)\) | \(e\left(\frac{4}{107}\right)\) | \(e\left(\frac{206}{321}\right)\) |
| \(\chi_{643}(38,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{107}\right)\) | \(e\left(\frac{49}{107}\right)\) | \(e\left(\frac{99}{107}\right)\) | \(e\left(\frac{77}{107}\right)\) | \(e\left(\frac{45}{107}\right)\) | \(e\left(\frac{302}{321}\right)\) | \(e\left(\frac{95}{107}\right)\) | \(e\left(\frac{98}{107}\right)\) | \(e\left(\frac{73}{107}\right)\) | \(e\left(\frac{175}{321}\right)\) |
| \(\chi_{643}(39,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{107}\right)\) | \(e\left(\frac{85}{107}\right)\) | \(e\left(\frac{56}{107}\right)\) | \(e\left(\frac{103}{107}\right)\) | \(e\left(\frac{6}{107}\right)\) | \(e\left(\frac{26}{321}\right)\) | \(e\left(\frac{84}{107}\right)\) | \(e\left(\frac{63}{107}\right)\) | \(e\left(\frac{24}{107}\right)\) | \(e\left(\frac{166}{321}\right)\) |
| \(\chi_{643}(42,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{107}\right)\) | \(e\left(\frac{91}{107}\right)\) | \(e\left(\frac{31}{107}\right)\) | \(e\left(\frac{36}{107}\right)\) | \(e\left(\frac{53}{107}\right)\) | \(e\left(\frac{194}{321}\right)\) | \(e\left(\frac{100}{107}\right)\) | \(e\left(\frac{75}{107}\right)\) | \(e\left(\frac{105}{107}\right)\) | \(e\left(\frac{4}{321}\right)\) |
| \(\chi_{643}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{107}\right)\) | \(e\left(\frac{20}{107}\right)\) | \(e\left(\frac{95}{107}\right)\) | \(e\left(\frac{62}{107}\right)\) | \(e\left(\frac{14}{107}\right)\) | \(e\left(\frac{25}{321}\right)\) | \(e\left(\frac{89}{107}\right)\) | \(e\left(\frac{40}{107}\right)\) | \(e\left(\frac{56}{107}\right)\) | \(e\left(\frac{209}{321}\right)\) |
| \(\chi_{643}(51,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{107}\right)\) | \(e\left(\frac{106}{107}\right)\) | \(e\left(\frac{22}{107}\right)\) | \(e\left(\frac{29}{107}\right)\) | \(e\left(\frac{10}{107}\right)\) | \(e\left(\frac{79}{321}\right)\) | \(e\left(\frac{33}{107}\right)\) | \(e\left(\frac{105}{107}\right)\) | \(e\left(\frac{40}{107}\right)\) | \(e\left(\frac{134}{321}\right)\) |
| \(\chi_{643}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{107}\right)\) | \(e\left(\frac{86}{107}\right)\) | \(e\left(\frac{34}{107}\right)\) | \(e\left(\frac{74}{107}\right)\) | \(e\left(\frac{103}{107}\right)\) | \(e\left(\frac{161}{321}\right)\) | \(e\left(\frac{51}{107}\right)\) | \(e\left(\frac{65}{107}\right)\) | \(e\left(\frac{91}{107}\right)\) | \(e\left(\frac{139}{321}\right)\) |
| \(\chi_{643}(55,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{107}\right)\) | \(e\left(\frac{51}{107}\right)\) | \(e\left(\frac{55}{107}\right)\) | \(e\left(\frac{19}{107}\right)\) | \(e\left(\frac{25}{107}\right)\) | \(e\left(\frac{37}{321}\right)\) | \(e\left(\frac{29}{107}\right)\) | \(e\left(\frac{102}{107}\right)\) | \(e\left(\frac{100}{107}\right)\) | \(e\left(\frac{14}{321}\right)\) |
| \(\chi_{643}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{107}\right)\) | \(e\left(\frac{16}{107}\right)\) | \(e\left(\frac{76}{107}\right)\) | \(e\left(\frac{71}{107}\right)\) | \(e\left(\frac{54}{107}\right)\) | \(e\left(\frac{20}{321}\right)\) | \(e\left(\frac{7}{107}\right)\) | \(e\left(\frac{32}{107}\right)\) | \(e\left(\frac{2}{107}\right)\) | \(e\left(\frac{103}{321}\right)\) |
| \(\chi_{643}(63,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{107}\right)\) | \(e\left(\frac{58}{107}\right)\) | \(e\left(\frac{8}{107}\right)\) | \(e\left(\frac{30}{107}\right)\) | \(e\left(\frac{62}{107}\right)\) | \(e\left(\frac{233}{321}\right)\) | \(e\left(\frac{12}{107}\right)\) | \(e\left(\frac{9}{107}\right)\) | \(e\left(\frac{34}{107}\right)\) | \(e\left(\frac{253}{321}\right)\) |
| \(\chi_{643}(65,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{30}{107}\right)\) | \(e\left(\frac{7}{107}\right)\) | \(e\left(\frac{60}{107}\right)\) | \(e\left(\frac{11}{107}\right)\) | \(e\left(\frac{37}{107}\right)\) | \(e\left(\frac{89}{321}\right)\) | \(e\left(\frac{90}{107}\right)\) | \(e\left(\frac{14}{107}\right)\) | \(e\left(\frac{41}{107}\right)\) | \(e\left(\frac{25}{321}\right)\) |
| \(\chi_{643}(70,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{107}\right)\) | \(e\left(\frac{13}{107}\right)\) | \(e\left(\frac{35}{107}\right)\) | \(e\left(\frac{51}{107}\right)\) | \(e\left(\frac{84}{107}\right)\) | \(e\left(\frac{257}{321}\right)\) | \(e\left(\frac{106}{107}\right)\) | \(e\left(\frac{26}{107}\right)\) | \(e\left(\frac{15}{107}\right)\) | \(e\left(\frac{184}{321}\right)\) |
| \(\chi_{643}(74,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{107}\right)\) | \(e\left(\frac{42}{107}\right)\) | \(e\left(\frac{39}{107}\right)\) | \(e\left(\frac{66}{107}\right)\) | \(e\left(\frac{8}{107}\right)\) | \(e\left(\frac{106}{321}\right)\) | \(e\left(\frac{5}{107}\right)\) | \(e\left(\frac{84}{107}\right)\) | \(e\left(\frac{32}{107}\right)\) | \(e\left(\frac{257}{321}\right)\) |
| \(\chi_{643}(82,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{107}\right)\) | \(e\left(\frac{104}{107}\right)\) | \(e\left(\frac{66}{107}\right)\) | \(e\left(\frac{87}{107}\right)\) | \(e\left(\frac{30}{107}\right)\) | \(e\left(\frac{23}{321}\right)\) | \(e\left(\frac{99}{107}\right)\) | \(e\left(\frac{101}{107}\right)\) | \(e\left(\frac{13}{107}\right)\) | \(e\left(\frac{295}{321}\right)\) |
| \(\chi_{643}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{107}\right)\) | \(e\left(\frac{77}{107}\right)\) | \(e\left(\frac{18}{107}\right)\) | \(e\left(\frac{14}{107}\right)\) | \(e\left(\frac{86}{107}\right)\) | \(e\left(\frac{230}{321}\right)\) | \(e\left(\frac{27}{107}\right)\) | \(e\left(\frac{47}{107}\right)\) | \(e\left(\frac{23}{107}\right)\) | \(e\left(\frac{61}{321}\right)\) |
| \(\chi_{643}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{107}\right)\) | \(e\left(\frac{28}{107}\right)\) | \(e\left(\frac{26}{107}\right)\) | \(e\left(\frac{44}{107}\right)\) | \(e\left(\frac{41}{107}\right)\) | \(e\left(\frac{142}{321}\right)\) | \(e\left(\frac{39}{107}\right)\) | \(e\left(\frac{56}{107}\right)\) | \(e\left(\frac{57}{107}\right)\) | \(e\left(\frac{314}{321}\right)\) |
| \(\chi_{643}(88,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{107}\right)\) | \(e\left(\frac{62}{107}\right)\) | \(e\left(\frac{27}{107}\right)\) | \(e\left(\frac{21}{107}\right)\) | \(e\left(\frac{22}{107}\right)\) | \(e\left(\frac{238}{321}\right)\) | \(e\left(\frac{94}{107}\right)\) | \(e\left(\frac{17}{107}\right)\) | \(e\left(\frac{88}{107}\right)\) | \(e\left(\frac{38}{321}\right)\) |
| \(\chi_{643}(89,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{92}{107}\right)\) | \(e\left(\frac{50}{107}\right)\) | \(e\left(\frac{77}{107}\right)\) | \(e\left(\frac{48}{107}\right)\) | \(e\left(\frac{35}{107}\right)\) | \(e\left(\frac{223}{321}\right)\) | \(e\left(\frac{62}{107}\right)\) | \(e\left(\frac{100}{107}\right)\) | \(e\left(\frac{33}{107}\right)\) | \(e\left(\frac{41}{321}\right)\) |
| \(\chi_{643}(92,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{107}\right)\) | \(e\left(\frac{97}{107}\right)\) | \(e\left(\frac{6}{107}\right)\) | \(e\left(\frac{76}{107}\right)\) | \(e\left(\frac{100}{107}\right)\) | \(e\left(\frac{148}{321}\right)\) | \(e\left(\frac{9}{107}\right)\) | \(e\left(\frac{87}{107}\right)\) | \(e\left(\frac{79}{107}\right)\) | \(e\left(\frac{56}{321}\right)\) |
| \(\chi_{643}(94,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{107}\right)\) | \(e\left(\frac{67}{107}\right)\) | \(e\left(\frac{24}{107}\right)\) | \(e\left(\frac{90}{107}\right)\) | \(e\left(\frac{79}{107}\right)\) | \(e\left(\frac{271}{321}\right)\) | \(e\left(\frac{36}{107}\right)\) | \(e\left(\frac{27}{107}\right)\) | \(e\left(\frac{102}{107}\right)\) | \(e\left(\frac{224}{321}\right)\) |
| \(\chi_{643}(95,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{107}\right)\) | \(e\left(\frac{45}{107}\right)\) | \(e\left(\frac{80}{107}\right)\) | \(e\left(\frac{86}{107}\right)\) | \(e\left(\frac{85}{107}\right)\) | \(e\left(\frac{83}{321}\right)\) | \(e\left(\frac{13}{107}\right)\) | \(e\left(\frac{90}{107}\right)\) | \(e\left(\frac{19}{107}\right)\) | \(e\left(\frac{283}{321}\right)\) |
| \(\chi_{643}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{107}\right)\) | \(e\left(\frac{45}{107}\right)\) | \(e\left(\frac{80}{107}\right)\) | \(e\left(\frac{86}{107}\right)\) | \(e\left(\frac{85}{107}\right)\) | \(e\left(\frac{190}{321}\right)\) | \(e\left(\frac{13}{107}\right)\) | \(e\left(\frac{90}{107}\right)\) | \(e\left(\frac{19}{107}\right)\) | \(e\left(\frac{176}{321}\right)\) |
| \(\chi_{643}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{107}\right)\) | \(e\left(\frac{11}{107}\right)\) | \(e\left(\frac{79}{107}\right)\) | \(e\left(\frac{2}{107}\right)\) | \(e\left(\frac{104}{107}\right)\) | \(e\left(\frac{94}{321}\right)\) | \(e\left(\frac{65}{107}\right)\) | \(e\left(\frac{22}{107}\right)\) | \(e\left(\frac{95}{107}\right)\) | \(e\left(\frac{131}{321}\right)\) |