from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(421, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,421))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(421\) | |
| Conductor: | \(421\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(420\) | 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: | $\Q(\zeta_{420})$ |
| Fixed field: | Number field defined by a degree 420 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{421}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{420}\right)\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{1}{140}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{93}{140}\right)\) | \(e\left(\frac{94}{105}\right)\) |
| \(\chi_{421}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{367}{420}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{193}{210}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{87}{140}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{111}{140}\right)\) | \(e\left(\frac{58}{105}\right)\) |
| \(\chi_{421}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{389}{420}\right)\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{179}{210}\right)\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{109}{140}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{57}{140}\right)\) | \(e\left(\frac{26}{105}\right)\) |
| \(\chi_{421}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{377}{420}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{167}{210}\right)\) | \(e\left(\frac{113}{210}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{97}{140}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{61}{140}\right)\) | \(e\left(\frac{53}{105}\right)\) |
| \(\chi_{421}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{187}{420}\right)\) | \(e\left(\frac{92}{105}\right)\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{163}{210}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{47}{140}\right)\) | \(e\left(\frac{79}{105}\right)\) | \(e\left(\frac{31}{140}\right)\) | \(e\left(\frac{43}{105}\right)\) |
| \(\chi_{421}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{407}{420}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{197}{210}\right)\) | \(e\left(\frac{83}{210}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{127}{140}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{51}{140}\right)\) | \(e\left(\frac{38}{105}\right)\) |
| \(\chi_{421}(30,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{263}{420}\right)\) | \(e\left(\frac{103}{105}\right)\) | \(e\left(\frac{53}{210}\right)\) | \(e\left(\frac{17}{210}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{123}{140}\right)\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{99}{140}\right)\) | \(e\left(\frac{47}{105}\right)\) |
| \(\chi_{421}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{420}\right)\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{47}{210}\right)\) | \(e\left(\frac{23}{210}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{47}{140}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{31}{140}\right)\) | \(e\left(\frac{8}{105}\right)\) |
| \(\chi_{421}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{281}{420}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{1}{140}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{93}{140}\right)\) | \(e\left(\frac{59}{105}\right)\) |
| \(\chi_{421}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{307}{420}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{97}{210}\right)\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{27}{140}\right)\) | \(e\left(\frac{64}{105}\right)\) | \(e\left(\frac{131}{140}\right)\) | \(e\left(\frac{88}{105}\right)\) |
| \(\chi_{421}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{420}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{109}{210}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{109}{140}\right)\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{57}{140}\right)\) | \(e\left(\frac{61}{105}\right)\) |
| \(\chi_{421}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{420}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{137}{210}\right)\) | \(e\left(\frac{143}{210}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{137}{140}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{1}{140}\right)\) | \(e\left(\frac{68}{105}\right)\) |
| \(\chi_{421}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{401}{420}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{191}{210}\right)\) | \(e\left(\frac{89}{210}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{53}{140}\right)\) | \(e\left(\frac{104}{105}\right)\) |
| \(\chi_{421}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{373}{420}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{163}{210}\right)\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{93}{140}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{109}{140}\right)\) | \(e\left(\frac{97}{105}\right)\) |
| \(\chi_{421}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{420}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{11}{210}\right)\) | \(e\left(\frac{59}{210}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{81}{140}\right)\) | \(e\left(\frac{17}{105}\right)\) | \(e\left(\frac{113}{140}\right)\) | \(e\left(\frac{89}{105}\right)\) |
| \(\chi_{421}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{341}{420}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{131}{210}\right)\) | \(e\left(\frac{149}{210}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{61}{140}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{73}{140}\right)\) | \(e\left(\frac{29}{105}\right)\) |
| \(\chi_{421}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{361}{420}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{199}{210}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{81}{140}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{113}{140}\right)\) | \(e\left(\frac{19}{105}\right)\) |
| \(\chi_{421}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{293}{420}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{83}{210}\right)\) | \(e\left(\frac{197}{210}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{13}{140}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{89}{140}\right)\) | \(e\left(\frac{32}{105}\right)\) |
| \(\chi_{421}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{391}{420}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{181}{210}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{111}{140}\right)\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{103}{140}\right)\) | \(e\left(\frac{4}{105}\right)\) |
| \(\chi_{421}(76,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{239}{420}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{29}{210}\right)\) | \(e\left(\frac{41}{210}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{99}{140}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{107}{140}\right)\) | \(e\left(\frac{101}{105}\right)\) |
| \(\chi_{421}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{337}{420}\right)\) | \(e\left(\frac{17}{105}\right)\) | \(e\left(\frac{127}{210}\right)\) | \(e\left(\frac{13}{210}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{57}{140}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{73}{105}\right)\) |
| \(\chi_{421}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{420}\right)\) | \(e\left(\frac{64}{105}\right)\) | \(e\left(\frac{89}{210}\right)\) | \(e\left(\frac{191}{210}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{89}{140}\right)\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{17}{140}\right)\) | \(e\left(\frac{71}{105}\right)\) |
| \(\chi_{421}(88,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{379}{420}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{181}{210}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{99}{140}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{107}{140}\right)\) | \(e\left(\frac{31}{105}\right)\) |
| \(\chi_{421}(90,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{247}{420}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{103}{210}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{107}{140}\right)\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{11}{140}\right)\) | \(e\left(\frac{13}{105}\right)\) |
| \(\chi_{421}(96,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{409}{420}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{199}{210}\right)\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{129}{140}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{97}{140}\right)\) | \(e\left(\frac{16}{105}\right)\) |
| \(\chi_{421}(98,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{313}{420}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{103}{210}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{33}{140}\right)\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{129}{140}\right)\) | \(e\left(\frac{22}{105}\right)\) |
| \(\chi_{421}(102,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{271}{420}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{61}{210}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{131}{140}\right)\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{3}{140}\right)\) | \(e\left(\frac{64}{105}\right)\) |
| \(\chi_{421}(116,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{420}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{107}{210}\right)\) | \(e\left(\frac{173}{210}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{107}{140}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{11}{140}\right)\) | \(e\left(\frac{83}{105}\right)\) |
| \(\chi_{421}(117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{420}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{109}{210}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{31}{140}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{83}{140}\right)\) | \(e\left(\frac{79}{105}\right)\) |
| \(\chi_{421}(134,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{420}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{97}{210}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{43}{140}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{79}{140}\right)\) | \(e\left(\frac{52}{105}\right)\) |
| \(\chi_{421}(136,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{289}{420}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{61}{210}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{9}{140}\right)\) | \(e\left(\frac{103}{105}\right)\) | \(e\left(\frac{137}{140}\right)\) | \(e\left(\frac{76}{105}\right)\) |