from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3584, base_ring=CyclotomicField(384))
M = H._module
chi = DirichletCharacter(H, M([192,255,256]))
chi.galois_orbit()
[g,chi] = znchar(Mod(11,3584))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3584\) | |
| Conductor: | \(3584\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(384\) | 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: | $\Q(\zeta_{384})$ |
| Fixed field: | Number field defined by a degree 384 polynomial (not computed) |
First 31 of 128 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3584}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{384}\right)\) | \(e\left(\frac{383}{384}\right)\) | \(e\left(\frac{157}{192}\right)\) | \(e\left(\frac{235}{384}\right)\) | \(e\left(\frac{91}{128}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{41}{384}\right)\) | \(e\left(\frac{25}{192}\right)\) | \(e\left(\frac{191}{192}\right)\) |
| \(\chi_{3584}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{384}\right)\) | \(e\left(\frac{61}{384}\right)\) | \(e\left(\frac{23}{192}\right)\) | \(e\left(\frac{257}{384}\right)\) | \(e\left(\frac{81}{128}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{187}{384}\right)\) | \(e\left(\frac{11}{192}\right)\) | \(e\left(\frac{61}{192}\right)\) |
| \(\chi_{3584}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{235}{384}\right)\) | \(e\left(\frac{89}{384}\right)\) | \(e\left(\frac{43}{192}\right)\) | \(e\left(\frac{205}{384}\right)\) | \(e\left(\frac{93}{128}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{191}{384}\right)\) | \(e\left(\frac{79}{192}\right)\) | \(e\left(\frac{89}{192}\right)\) |
| \(\chi_{3584}(107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{384}\right)\) | \(e\left(\frac{295}{384}\right)\) | \(e\left(\frac{149}{192}\right)\) | \(e\left(\frac{179}{384}\right)\) | \(e\left(\frac{35}{128}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{193}{384}\right)\) | \(e\left(\frac{113}{192}\right)\) | \(e\left(\frac{103}{192}\right)\) |
| \(\chi_{3584}(123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{384}\right)\) | \(e\left(\frac{227}{384}\right)\) | \(e\left(\frac{73}{192}\right)\) | \(e\left(\frac{31}{384}\right)\) | \(e\left(\frac{79}{128}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{293}{384}\right)\) | \(e\left(\frac{85}{192}\right)\) | \(e\left(\frac{35}{192}\right)\) |
| \(\chi_{3584}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{384}\right)\) | \(e\left(\frac{193}{384}\right)\) | \(e\left(\frac{35}{192}\right)\) | \(e\left(\frac{341}{384}\right)\) | \(e\left(\frac{101}{128}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{151}{384}\right)\) | \(e\left(\frac{167}{192}\right)\) | \(e\left(\frac{1}{192}\right)\) |
| \(\chi_{3584}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{384}\right)\) | \(e\left(\frac{29}{384}\right)\) | \(e\left(\frac{55}{192}\right)\) | \(e\left(\frac{97}{384}\right)\) | \(e\left(\frac{49}{128}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{347}{384}\right)\) | \(e\left(\frac{43}{192}\right)\) | \(e\left(\frac{29}{192}\right)\) |
| \(\chi_{3584}(219,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{384}\right)\) | \(e\left(\frac{139}{384}\right)\) | \(e\left(\frac{65}{192}\right)\) | \(e\left(\frac{359}{384}\right)\) | \(e\left(\frac{23}{128}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{61}{384}\right)\) | \(e\left(\frac{173}{192}\right)\) | \(e\left(\frac{139}{192}\right)\) |
| \(\chi_{3584}(235,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{384}\right)\) | \(e\left(\frac{263}{384}\right)\) | \(e\left(\frac{181}{192}\right)\) | \(e\left(\frac{19}{384}\right)\) | \(e\left(\frac{3}{128}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{353}{384}\right)\) | \(e\left(\frac{145}{192}\right)\) | \(e\left(\frac{71}{192}\right)\) |
| \(\chi_{3584}(275,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{239}{384}\right)\) | \(e\left(\frac{133}{384}\right)\) | \(e\left(\frac{47}{192}\right)\) | \(e\left(\frac{233}{384}\right)\) | \(e\left(\frac{57}{128}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{307}{384}\right)\) | \(e\left(\frac{131}{192}\right)\) | \(e\left(\frac{133}{192}\right)\) |
| \(\chi_{3584}(291,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{384}\right)\) | \(e\left(\frac{161}{384}\right)\) | \(e\left(\frac{67}{192}\right)\) | \(e\left(\frac{181}{384}\right)\) | \(e\left(\frac{69}{128}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{311}{384}\right)\) | \(e\left(\frac{7}{192}\right)\) | \(e\left(\frac{161}{192}\right)\) |
| \(\chi_{3584}(331,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{173}{384}\right)\) | \(e\left(\frac{175}{384}\right)\) | \(e\left(\frac{173}{192}\right)\) | \(e\left(\frac{347}{384}\right)\) | \(e\left(\frac{75}{128}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{121}{384}\right)\) | \(e\left(\frac{41}{192}\right)\) | \(e\left(\frac{175}{192}\right)\) |
| \(\chi_{3584}(347,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{384}\right)\) | \(e\left(\frac{107}{384}\right)\) | \(e\left(\frac{97}{192}\right)\) | \(e\left(\frac{199}{384}\right)\) | \(e\left(\frac{119}{128}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{221}{384}\right)\) | \(e\left(\frac{13}{192}\right)\) | \(e\left(\frac{107}{192}\right)\) |
| \(\chi_{3584}(387,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{251}{384}\right)\) | \(e\left(\frac{265}{384}\right)\) | \(e\left(\frac{59}{192}\right)\) | \(e\left(\frac{317}{384}\right)\) | \(e\left(\frac{77}{128}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{271}{384}\right)\) | \(e\left(\frac{95}{192}\right)\) | \(e\left(\frac{73}{192}\right)\) |
| \(\chi_{3584}(403,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{271}{384}\right)\) | \(e\left(\frac{101}{384}\right)\) | \(e\left(\frac{79}{192}\right)\) | \(e\left(\frac{73}{384}\right)\) | \(e\left(\frac{25}{128}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{83}{384}\right)\) | \(e\left(\frac{163}{192}\right)\) | \(e\left(\frac{101}{192}\right)\) |
| \(\chi_{3584}(443,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{384}\right)\) | \(e\left(\frac{19}{384}\right)\) | \(e\left(\frac{89}{192}\right)\) | \(e\left(\frac{143}{384}\right)\) | \(e\left(\frac{63}{128}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{373}{384}\right)\) | \(e\left(\frac{101}{192}\right)\) | \(e\left(\frac{19}{192}\right)\) |
| \(\chi_{3584}(459,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{205}{384}\right)\) | \(e\left(\frac{143}{384}\right)\) | \(e\left(\frac{13}{192}\right)\) | \(e\left(\frac{187}{384}\right)\) | \(e\left(\frac{43}{128}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{281}{384}\right)\) | \(e\left(\frac{73}{192}\right)\) | \(e\left(\frac{143}{192}\right)\) |
| \(\chi_{3584}(499,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{384}\right)\) | \(e\left(\frac{205}{384}\right)\) | \(e\left(\frac{71}{192}\right)\) | \(e\left(\frac{209}{384}\right)\) | \(e\left(\frac{33}{128}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{43}{384}\right)\) | \(e\left(\frac{59}{192}\right)\) | \(e\left(\frac{13}{192}\right)\) |
| \(\chi_{3584}(515,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{283}{384}\right)\) | \(e\left(\frac{233}{384}\right)\) | \(e\left(\frac{91}{192}\right)\) | \(e\left(\frac{157}{384}\right)\) | \(e\left(\frac{45}{128}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{31}{96}\right)\) | \(e\left(\frac{47}{384}\right)\) | \(e\left(\frac{127}{192}\right)\) | \(e\left(\frac{41}{192}\right)\) |
| \(\chi_{3584}(555,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{197}{384}\right)\) | \(e\left(\frac{55}{384}\right)\) | \(e\left(\frac{5}{192}\right)\) | \(e\left(\frac{131}{384}\right)\) | \(e\left(\frac{115}{128}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{49}{384}\right)\) | \(e\left(\frac{161}{192}\right)\) | \(e\left(\frac{55}{192}\right)\) |
| \(\chi_{3584}(571,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{384}\right)\) | \(e\left(\frac{371}{384}\right)\) | \(e\left(\frac{121}{192}\right)\) | \(e\left(\frac{367}{384}\right)\) | \(e\left(\frac{31}{128}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{149}{384}\right)\) | \(e\left(\frac{133}{192}\right)\) | \(e\left(\frac{179}{192}\right)\) |
| \(\chi_{3584}(611,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{384}\right)\) | \(e\left(\frac{337}{384}\right)\) | \(e\left(\frac{83}{192}\right)\) | \(e\left(\frac{293}{384}\right)\) | \(e\left(\frac{53}{128}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{7}{384}\right)\) | \(e\left(\frac{23}{192}\right)\) | \(e\left(\frac{145}{192}\right)\) |
| \(\chi_{3584}(627,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{384}\right)\) | \(e\left(\frac{173}{384}\right)\) | \(e\left(\frac{103}{192}\right)\) | \(e\left(\frac{49}{384}\right)\) | \(e\left(\frac{1}{128}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{203}{384}\right)\) | \(e\left(\frac{91}{192}\right)\) | \(e\left(\frac{173}{192}\right)\) |
| \(\chi_{3584}(667,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{384}\right)\) | \(e\left(\frac{283}{384}\right)\) | \(e\left(\frac{113}{192}\right)\) | \(e\left(\frac{311}{384}\right)\) | \(e\left(\frac{103}{128}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{301}{384}\right)\) | \(e\left(\frac{29}{192}\right)\) | \(e\left(\frac{91}{192}\right)\) |
| \(\chi_{3584}(683,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{229}{384}\right)\) | \(e\left(\frac{23}{384}\right)\) | \(e\left(\frac{37}{192}\right)\) | \(e\left(\frac{355}{384}\right)\) | \(e\left(\frac{83}{128}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{209}{384}\right)\) | \(e\left(\frac{1}{192}\right)\) | \(e\left(\frac{23}{192}\right)\) |
| \(\chi_{3584}(723,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{287}{384}\right)\) | \(e\left(\frac{277}{384}\right)\) | \(e\left(\frac{95}{192}\right)\) | \(e\left(\frac{185}{384}\right)\) | \(e\left(\frac{9}{128}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{163}{384}\right)\) | \(e\left(\frac{179}{192}\right)\) | \(e\left(\frac{85}{192}\right)\) |
| \(\chi_{3584}(739,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{384}\right)\) | \(e\left(\frac{305}{384}\right)\) | \(e\left(\frac{115}{192}\right)\) | \(e\left(\frac{133}{384}\right)\) | \(e\left(\frac{21}{128}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{167}{384}\right)\) | \(e\left(\frac{55}{192}\right)\) | \(e\left(\frac{113}{192}\right)\) |
| \(\chi_{3584}(779,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{384}\right)\) | \(e\left(\frac{319}{384}\right)\) | \(e\left(\frac{29}{192}\right)\) | \(e\left(\frac{299}{384}\right)\) | \(e\left(\frac{27}{128}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{361}{384}\right)\) | \(e\left(\frac{89}{192}\right)\) | \(e\left(\frac{127}{192}\right)\) |
| \(\chi_{3584}(795,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{384}\right)\) | \(e\left(\frac{251}{384}\right)\) | \(e\left(\frac{145}{192}\right)\) | \(e\left(\frac{151}{384}\right)\) | \(e\left(\frac{71}{128}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{77}{384}\right)\) | \(e\left(\frac{61}{192}\right)\) | \(e\left(\frac{59}{192}\right)\) |
| \(\chi_{3584}(835,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{299}{384}\right)\) | \(e\left(\frac{25}{384}\right)\) | \(e\left(\frac{107}{192}\right)\) | \(e\left(\frac{269}{384}\right)\) | \(e\left(\frac{29}{128}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{127}{384}\right)\) | \(e\left(\frac{143}{192}\right)\) | \(e\left(\frac{25}{192}\right)\) |
| \(\chi_{3584}(851,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{319}{384}\right)\) | \(e\left(\frac{245}{384}\right)\) | \(e\left(\frac{127}{192}\right)\) | \(e\left(\frac{25}{384}\right)\) | \(e\left(\frac{105}{128}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{323}{384}\right)\) | \(e\left(\frac{19}{192}\right)\) | \(e\left(\frac{53}{192}\right)\) |