from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(643, base_ring=CyclotomicField(642))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(11,643))
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: | \(642\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | 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 642 polynomial (not computed) |
First 31 of 212 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{643}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{214}\right)\) | \(e\left(\frac{103}{214}\right)\) | \(e\left(\frac{44}{107}\right)\) | \(e\left(\frac{9}{214}\right)\) | \(e\left(\frac{20}{107}\right)\) | \(e\left(\frac{265}{321}\right)\) | \(e\left(\frac{25}{214}\right)\) | \(e\left(\frac{103}{107}\right)\) | \(e\left(\frac{80}{107}\right)\) | \(e\left(\frac{1}{642}\right)\) |
| \(\chi_{643}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{214}\right)\) | \(e\left(\frac{15}{214}\right)\) | \(e\left(\frac{49}{107}\right)\) | \(e\left(\frac{207}{214}\right)\) | \(e\left(\frac{32}{107}\right)\) | \(e\left(\frac{317}{321}\right)\) | \(e\left(\frac{147}{214}\right)\) | \(e\left(\frac{15}{107}\right)\) | \(e\left(\frac{21}{107}\right)\) | \(e\left(\frac{23}{642}\right)\) |
| \(\chi_{643}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{214}\right)\) | \(e\left(\frac{27}{214}\right)\) | \(e\left(\frac{24}{107}\right)\) | \(e\left(\frac{73}{214}\right)\) | \(e\left(\frac{79}{107}\right)\) | \(e\left(\frac{164}{321}\right)\) | \(e\left(\frac{179}{214}\right)\) | \(e\left(\frac{27}{107}\right)\) | \(e\left(\frac{102}{107}\right)\) | \(e\left(\frac{341}{642}\right)\) |
| \(\chi_{643}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{214}\right)\) | \(e\left(\frac{57}{214}\right)\) | \(e\left(\frac{15}{107}\right)\) | \(e\left(\frac{59}{214}\right)\) | \(e\left(\frac{36}{107}\right)\) | \(e\left(\frac{49}{321}\right)\) | \(e\left(\frac{45}{214}\right)\) | \(e\left(\frac{57}{107}\right)\) | \(e\left(\frac{37}{107}\right)\) | \(e\left(\frac{601}{642}\right)\) |
| \(\chi_{643}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{214}\right)\) | \(e\left(\frac{91}{214}\right)\) | \(e\left(\frac{69}{107}\right)\) | \(e\left(\frac{143}{214}\right)\) | \(e\left(\frac{80}{107}\right)\) | \(e\left(\frac{311}{321}\right)\) | \(e\left(\frac{207}{214}\right)\) | \(e\left(\frac{91}{107}\right)\) | \(e\left(\frac{106}{107}\right)\) | \(e\left(\frac{539}{642}\right)\) |
| \(\chi_{643}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{214}\right)\) | \(e\left(\frac{175}{214}\right)\) | \(e\left(\frac{1}{107}\right)\) | \(e\left(\frac{61}{214}\right)\) | \(e\left(\frac{88}{107}\right)\) | \(e\left(\frac{203}{321}\right)\) | \(e\left(\frac{3}{214}\right)\) | \(e\left(\frac{68}{107}\right)\) | \(e\left(\frac{31}{107}\right)\) | \(e\left(\frac{197}{642}\right)\) |
| \(\chi_{643}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{214}\right)\) | \(e\left(\frac{19}{214}\right)\) | \(e\left(\frac{5}{107}\right)\) | \(e\left(\frac{91}{214}\right)\) | \(e\left(\frac{12}{107}\right)\) | \(e\left(\frac{266}{321}\right)\) | \(e\left(\frac{15}{214}\right)\) | \(e\left(\frac{19}{107}\right)\) | \(e\left(\frac{48}{107}\right)\) | \(e\left(\frac{557}{642}\right)\) |
| \(\chi_{643}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{214}\right)\) | \(e\left(\frac{77}{214}\right)\) | \(e\left(\frac{9}{107}\right)\) | \(e\left(\frac{121}{214}\right)\) | \(e\left(\frac{43}{107}\right)\) | \(e\left(\frac{115}{321}\right)\) | \(e\left(\frac{27}{214}\right)\) | \(e\left(\frac{77}{107}\right)\) | \(e\left(\frac{65}{107}\right)\) | \(e\left(\frac{61}{642}\right)\) |
| \(\chi_{643}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{214}\right)\) | \(e\left(\frac{201}{214}\right)\) | \(e\left(\frac{36}{107}\right)\) | \(e\left(\frac{163}{214}\right)\) | \(e\left(\frac{65}{107}\right)\) | \(e\left(\frac{32}{321}\right)\) | \(e\left(\frac{1}{214}\right)\) | \(e\left(\frac{94}{107}\right)\) | \(e\left(\frac{46}{107}\right)\) | \(e\left(\frac{137}{642}\right)\) |
| \(\chi_{643}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{211}{214}\right)\) | \(e\left(\frac{117}{214}\right)\) | \(e\left(\frac{104}{107}\right)\) | \(e\left(\frac{31}{214}\right)\) | \(e\left(\frac{57}{107}\right)\) | \(e\left(\frac{247}{321}\right)\) | \(e\left(\frac{205}{214}\right)\) | \(e\left(\frac{10}{107}\right)\) | \(e\left(\frac{14}{107}\right)\) | \(e\left(\frac{265}{642}\right)\) |
| \(\chi_{643}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{214}\right)\) | \(e\left(\frac{187}{214}\right)\) | \(e\left(\frac{83}{107}\right)\) | \(e\left(\frac{141}{214}\right)\) | \(e\left(\frac{28}{107}\right)\) | \(e\left(\frac{157}{321}\right)\) | \(e\left(\frac{35}{214}\right)\) | \(e\left(\frac{80}{107}\right)\) | \(e\left(\frac{5}{107}\right)\) | \(e\left(\frac{301}{642}\right)\) |
| \(\chi_{643}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{214}\right)\) | \(e\left(\frac{127}{214}\right)\) | \(e\left(\frac{101}{107}\right)\) | \(e\left(\frac{169}{214}\right)\) | \(e\left(\frac{7}{107}\right)\) | \(e\left(\frac{280}{321}\right)\) | \(e\left(\frac{89}{214}\right)\) | \(e\left(\frac{20}{107}\right)\) | \(e\left(\frac{28}{107}\right)\) | \(e\left(\frac{637}{642}\right)\) |
| \(\chi_{643}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{214}\right)\) | \(e\left(\frac{29}{214}\right)\) | \(e\left(\frac{2}{107}\right)\) | \(e\left(\frac{15}{214}\right)\) | \(e\left(\frac{69}{107}\right)\) | \(e\left(\frac{299}{321}\right)\) | \(e\left(\frac{113}{214}\right)\) | \(e\left(\frac{29}{107}\right)\) | \(e\left(\frac{62}{107}\right)\) | \(e\left(\frac{287}{642}\right)\) |
| \(\chi_{643}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{214}\right)\) | \(e\left(\frac{41}{214}\right)\) | \(e\left(\frac{84}{107}\right)\) | \(e\left(\frac{95}{214}\right)\) | \(e\left(\frac{9}{107}\right)\) | \(e\left(\frac{146}{321}\right)\) | \(e\left(\frac{145}{214}\right)\) | \(e\left(\frac{41}{107}\right)\) | \(e\left(\frac{36}{107}\right)\) | \(e\left(\frac{605}{642}\right)\) |
| \(\chi_{643}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{214}\right)\) | \(e\left(\frac{3}{214}\right)\) | \(e\left(\frac{74}{107}\right)\) | \(e\left(\frac{127}{214}\right)\) | \(e\left(\frac{92}{107}\right)\) | \(e\left(\frac{149}{321}\right)\) | \(e\left(\frac{115}{214}\right)\) | \(e\left(\frac{3}{107}\right)\) | \(e\left(\frac{47}{107}\right)\) | \(e\left(\frac{347}{642}\right)\) |
| \(\chi_{643}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{207}{214}\right)\) | \(e\left(\frac{59}{214}\right)\) | \(e\left(\frac{100}{107}\right)\) | \(e\left(\frac{1}{214}\right)\) | \(e\left(\frac{26}{107}\right)\) | \(e\left(\frac{77}{321}\right)\) | \(e\left(\frac{193}{214}\right)\) | \(e\left(\frac{59}{107}\right)\) | \(e\left(\frac{104}{107}\right)\) | \(e\left(\frac{119}{642}\right)\) |
| \(\chi_{643}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{117}{214}\right)\) | \(e\left(\frac{145}{214}\right)\) | \(e\left(\frac{10}{107}\right)\) | \(e\left(\frac{75}{214}\right)\) | \(e\left(\frac{24}{107}\right)\) | \(e\left(\frac{211}{321}\right)\) | \(e\left(\frac{137}{214}\right)\) | \(e\left(\frac{38}{107}\right)\) | \(e\left(\frac{96}{107}\right)\) | \(e\left(\frac{151}{642}\right)\) |
| \(\chi_{643}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{214}\right)\) | \(e\left(\frac{67}{214}\right)\) | \(e\left(\frac{12}{107}\right)\) | \(e\left(\frac{197}{214}\right)\) | \(e\left(\frac{93}{107}\right)\) | \(e\left(\frac{82}{321}\right)\) | \(e\left(\frac{143}{214}\right)\) | \(e\left(\frac{67}{107}\right)\) | \(e\left(\frac{51}{107}\right)\) | \(e\left(\frac{331}{642}\right)\) |
| \(\chi_{643}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{81}{214}\right)\) | \(e\left(\frac{51}{214}\right)\) | \(e\left(\frac{81}{107}\right)\) | \(e\left(\frac{19}{214}\right)\) | \(e\left(\frac{66}{107}\right)\) | \(e\left(\frac{286}{321}\right)\) | \(e\left(\frac{29}{214}\right)\) | \(e\left(\frac{51}{107}\right)\) | \(e\left(\frac{50}{107}\right)\) | \(e\left(\frac{121}{642}\right)\) |
| \(\chi_{643}(68,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{75}{214}\right)\) | \(e\left(\frac{71}{214}\right)\) | \(e\left(\frac{75}{107}\right)\) | \(e\left(\frac{81}{214}\right)\) | \(e\left(\frac{73}{107}\right)\) | \(e\left(\frac{31}{321}\right)\) | \(e\left(\frac{11}{214}\right)\) | \(e\left(\frac{71}{107}\right)\) | \(e\left(\frac{78}{107}\right)\) | \(e\left(\frac{223}{642}\right)\) |
| \(\chi_{643}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{214}\right)\) | \(e\left(\frac{121}{214}\right)\) | \(e\left(\frac{60}{107}\right)\) | \(e\left(\frac{129}{214}\right)\) | \(e\left(\frac{37}{107}\right)\) | \(e\left(\frac{196}{321}\right)\) | \(e\left(\frac{73}{214}\right)\) | \(e\left(\frac{14}{107}\right)\) | \(e\left(\frac{41}{107}\right)\) | \(e\left(\frac{157}{642}\right)\) |
| \(\chi_{643}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{117}{214}\right)\) | \(e\left(\frac{145}{214}\right)\) | \(e\left(\frac{10}{107}\right)\) | \(e\left(\frac{75}{214}\right)\) | \(e\left(\frac{24}{107}\right)\) | \(e\left(\frac{104}{321}\right)\) | \(e\left(\frac{137}{214}\right)\) | \(e\left(\frac{38}{107}\right)\) | \(e\left(\frac{96}{107}\right)\) | \(e\left(\frac{365}{642}\right)\) |
| \(\chi_{643}(76,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{129}{214}\right)\) | \(e\left(\frac{105}{214}\right)\) | \(e\left(\frac{22}{107}\right)\) | \(e\left(\frac{165}{214}\right)\) | \(e\left(\frac{10}{107}\right)\) | \(e\left(\frac{293}{321}\right)\) | \(e\left(\frac{173}{214}\right)\) | \(e\left(\frac{105}{107}\right)\) | \(e\left(\frac{40}{107}\right)\) | \(e\left(\frac{161}{642}\right)\) |
| \(\chi_{643}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{214}\right)\) | \(e\left(\frac{177}{214}\right)\) | \(e\left(\frac{86}{107}\right)\) | \(e\left(\frac{3}{214}\right)\) | \(e\left(\frac{78}{107}\right)\) | \(e\left(\frac{17}{321}\right)\) | \(e\left(\frac{151}{214}\right)\) | \(e\left(\frac{70}{107}\right)\) | \(e\left(\frac{98}{107}\right)\) | \(e\left(\frac{143}{642}\right)\) |
| \(\chi_{643}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{214}\right)\) | \(e\left(\frac{189}{214}\right)\) | \(e\left(\frac{61}{107}\right)\) | \(e\left(\frac{83}{214}\right)\) | \(e\left(\frac{18}{107}\right)\) | \(e\left(\frac{292}{321}\right)\) | \(e\left(\frac{183}{214}\right)\) | \(e\left(\frac{82}{107}\right)\) | \(e\left(\frac{72}{107}\right)\) | \(e\left(\frac{247}{642}\right)\) |
| \(\chi_{643}(84,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{214}\right)\) | \(e\left(\frac{189}{214}\right)\) | \(e\left(\frac{61}{107}\right)\) | \(e\left(\frac{83}{214}\right)\) | \(e\left(\frac{18}{107}\right)\) | \(e\left(\frac{185}{321}\right)\) | \(e\left(\frac{183}{214}\right)\) | \(e\left(\frac{82}{107}\right)\) | \(e\left(\frac{72}{107}\right)\) | \(e\left(\frac{461}{642}\right)\) |
| \(\chi_{643}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{214}\right)\) | \(e\left(\frac{151}{214}\right)\) | \(e\left(\frac{51}{107}\right)\) | \(e\left(\frac{115}{214}\right)\) | \(e\left(\frac{101}{107}\right)\) | \(e\left(\frac{188}{321}\right)\) | \(e\left(\frac{153}{214}\right)\) | \(e\left(\frac{44}{107}\right)\) | \(e\left(\frac{83}{107}\right)\) | \(e\left(\frac{203}{642}\right)\) |
| \(\chi_{643}(91,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{214}\right)\) | \(e\left(\frac{35}{214}\right)\) | \(e\left(\frac{43}{107}\right)\) | \(e\left(\frac{55}{214}\right)\) | \(e\left(\frac{39}{107}\right)\) | \(e\left(\frac{169}{321}\right)\) | \(e\left(\frac{129}{214}\right)\) | \(e\left(\frac{35}{107}\right)\) | \(e\left(\frac{49}{107}\right)\) | \(e\left(\frac{553}{642}\right)\) |
| \(\chi_{643}(93,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{203}{214}\right)\) | \(e\left(\frac{1}{214}\right)\) | \(e\left(\frac{96}{107}\right)\) | \(e\left(\frac{185}{214}\right)\) | \(e\left(\frac{102}{107}\right)\) | \(e\left(\frac{121}{321}\right)\) | \(e\left(\frac{181}{214}\right)\) | \(e\left(\frac{1}{107}\right)\) | \(e\left(\frac{87}{107}\right)\) | \(e\left(\frac{187}{642}\right)\) |
| \(\chi_{643}(98,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{214}\right)\) | \(e\left(\frac{47}{214}\right)\) | \(e\left(\frac{18}{107}\right)\) | \(e\left(\frac{135}{214}\right)\) | \(e\left(\frac{86}{107}\right)\) | \(e\left(\frac{16}{321}\right)\) | \(e\left(\frac{161}{214}\right)\) | \(e\left(\frac{47}{107}\right)\) | \(e\left(\frac{23}{107}\right)\) | \(e\left(\frac{229}{642}\right)\) |
| \(\chi_{643}(99,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{165}{214}\right)\) | \(e\left(\frac{199}{214}\right)\) | \(e\left(\frac{58}{107}\right)\) | \(e\left(\frac{7}{214}\right)\) | \(e\left(\frac{75}{107}\right)\) | \(e\left(\frac{4}{321}\right)\) | \(e\left(\frac{67}{214}\right)\) | \(e\left(\frac{92}{107}\right)\) | \(e\left(\frac{86}{107}\right)\) | \(e\left(\frac{619}{642}\right)\) |