from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8624, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([0,105,290,168]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,8624))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8624\) | |
| Conductor: | \(8624\) | 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8624}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{269}{420}\right)\) | \(e\left(\frac{367}{420}\right)\) | \(e\left(\frac{59}{210}\right)\) | \(e\left(\frac{131}{140}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{181}{210}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{129}{140}\right)\) |
| \(\chi_{8624}(157,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{403}{420}\right)\) | \(e\left(\frac{389}{420}\right)\) | \(e\left(\frac{193}{210}\right)\) | \(e\left(\frac{37}{140}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{197}{210}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{179}{210}\right)\) | \(e\left(\frac{123}{140}\right)\) |
| \(\chi_{8624}(213,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{397}{420}\right)\) | \(e\left(\frac{131}{420}\right)\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{83}{140}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{143}{210}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{131}{210}\right)\) | \(e\left(\frac{117}{140}\right)\) |
| \(\chi_{8624}(229,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{420}\right)\) | \(e\left(\frac{403}{420}\right)\) | \(e\left(\frac{11}{210}\right)\) | \(e\left(\frac{79}{140}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{193}{210}\right)\) | \(e\left(\frac{81}{140}\right)\) |
| \(\chi_{8624}(269,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{420}\right)\) | \(e\left(\frac{377}{420}\right)\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{101}{140}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{131}{210}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{167}{210}\right)\) | \(e\left(\frac{139}{140}\right)\) |
| \(\chi_{8624}(493,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{420}\right)\) | \(e\left(\frac{353}{420}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{89}{140}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{143}{210}\right)\) | \(e\left(\frac{31}{140}\right)\) |
| \(\chi_{8624}(565,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{317}{420}\right)\) | \(e\left(\frac{331}{420}\right)\) | \(e\left(\frac{107}{210}\right)\) | \(e\left(\frac{43}{140}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{193}{210}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{121}{210}\right)\) | \(e\left(\frac{37}{140}\right)\) |
| \(\chi_{8624}(621,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{420}\right)\) | \(e\left(\frac{277}{420}\right)\) | \(e\left(\frac{179}{210}\right)\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{67}{210}\right)\) | \(e\left(\frac{39}{140}\right)\) |
| \(\chi_{8624}(773,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{253}{420}\right)\) | \(e\left(\frac{239}{420}\right)\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{67}{140}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{107}{210}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{29}{210}\right)\) | \(e\left(\frac{113}{140}\right)\) |
| \(\chi_{8624}(829,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{247}{420}\right)\) | \(e\left(\frac{401}{420}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{113}{140}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{53}{210}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{191}{210}\right)\) | \(e\left(\frac{107}{140}\right)\) |
| \(\chi_{8624}(845,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{420}\right)\) | \(e\left(\frac{313}{420}\right)\) | \(e\left(\frac{131}{210}\right)\) | \(e\left(\frac{69}{140}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{199}{210}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{103}{210}\right)\) | \(e\left(\frac{131}{140}\right)\) |
| \(\chi_{8624}(885,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{409}{420}\right)\) | \(e\left(\frac{227}{420}\right)\) | \(e\left(\frac{199}{210}\right)\) | \(e\left(\frac{131}{140}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{41}{210}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{17}{210}\right)\) | \(e\left(\frac{129}{140}\right)\) |
| \(\chi_{8624}(1125,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{420}\right)\) | \(e\left(\frac{379}{420}\right)\) | \(e\left(\frac{113}{210}\right)\) | \(e\left(\frac{67}{140}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{113}{140}\right)\) |
| \(\chi_{8624}(1181,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{420}\right)\) | \(e\left(\frac{241}{420}\right)\) | \(e\left(\frac{17}{210}\right)\) | \(e\left(\frac{33}{140}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{13}{210}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{87}{140}\right)\) |
| \(\chi_{8624}(1237,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{420}\right)\) | \(e\left(\frac{187}{420}\right)\) | \(e\left(\frac{89}{210}\right)\) | \(e\left(\frac{111}{140}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{89}{140}\right)\) |
| \(\chi_{8624}(1389,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{420}\right)\) | \(e\left(\frac{89}{420}\right)\) | \(e\left(\frac{103}{210}\right)\) | \(e\left(\frac{97}{140}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{17}{210}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{89}{210}\right)\) | \(e\left(\frac{103}{140}\right)\) |
| \(\chi_{8624}(1445,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{420}\right)\) | \(e\left(\frac{251}{420}\right)\) | \(e\left(\frac{97}{210}\right)\) | \(e\left(\frac{3}{140}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{173}{210}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{41}{210}\right)\) | \(e\left(\frac{97}{140}\right)\) |
| \(\chi_{8624}(1461,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{420}\right)\) | \(e\left(\frac{223}{420}\right)\) | \(e\left(\frac{41}{210}\right)\) | \(e\left(\frac{59}{140}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{19}{210}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{13}{210}\right)\) | \(e\left(\frac{41}{140}\right)\) |
| \(\chi_{8624}(1725,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{420}\right)\) | \(e\left(\frac{53}{420}\right)\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{9}{140}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{29}{210}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{53}{210}\right)\) | \(e\left(\frac{11}{140}\right)\) |
| \(\chi_{8624}(1741,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{420}\right)\) | \(e\left(\frac{289}{420}\right)\) | \(e\left(\frac{23}{210}\right)\) | \(e\left(\frac{57}{140}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{67}{210}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{23}{140}\right)\) |
| \(\chi_{8624}(1797,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{420}\right)\) | \(e\left(\frac{151}{420}\right)\) | \(e\left(\frac{137}{210}\right)\) | \(e\left(\frac{23}{140}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{137}{140}\right)\) |
| \(\chi_{8624}(1853,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{419}{420}\right)\) | \(e\left(\frac{97}{420}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{101}{140}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{61}{210}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{97}{210}\right)\) | \(e\left(\frac{139}{140}\right)\) |
| \(\chi_{8624}(2005,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{373}{420}\right)\) | \(e\left(\frac{359}{420}\right)\) | \(e\left(\frac{163}{210}\right)\) | \(e\left(\frac{127}{140}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{137}{210}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{149}{210}\right)\) | \(e\left(\frac{93}{140}\right)\) |
| \(\chi_{8624}(2061,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{367}{420}\right)\) | \(e\left(\frac{101}{420}\right)\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{33}{140}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{83}{210}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{87}{140}\right)\) |
| \(\chi_{8624}(2117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{420}\right)\) | \(e\left(\frac{347}{420}\right)\) | \(e\left(\frac{109}{210}\right)\) | \(e\left(\frac{51}{140}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{137}{210}\right)\) | \(e\left(\frac{109}{140}\right)\) |
| \(\chi_{8624}(2341,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{420}\right)\) | \(e\left(\frac{323}{420}\right)\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{39}{140}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{149}{210}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{113}{210}\right)\) | \(e\left(\frac{1}{140}\right)\) |
| \(\chi_{8624}(2357,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{353}{420}\right)\) | \(e\left(\frac{199}{420}\right)\) | \(e\left(\frac{143}{210}\right)\) | \(e\left(\frac{47}{140}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{97}{210}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{199}{210}\right)\) | \(e\left(\frac{73}{140}\right)\) |
| \(\chi_{8624}(2413,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{420}\right)\) | \(e\left(\frac{61}{420}\right)\) | \(e\left(\frac{47}{210}\right)\) | \(e\left(\frac{13}{140}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{73}{210}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{61}{210}\right)\) | \(e\left(\frac{47}{140}\right)\) |
| \(\chi_{8624}(2621,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{223}{420}\right)\) | \(e\left(\frac{209}{420}\right)\) | \(e\left(\frac{13}{210}\right)\) | \(e\left(\frac{17}{140}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{47}{210}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{83}{140}\right)\) |
| \(\chi_{8624}(2693,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{281}{420}\right)\) | \(e\left(\frac{43}{420}\right)\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{39}{140}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{1}{140}\right)\) |
| \(\chi_{8624}(2733,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{379}{420}\right)\) | \(e\left(\frac{197}{420}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{81}{140}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{191}{210}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{197}{210}\right)\) | \(e\left(\frac{99}{140}\right)\) |