from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4608, base_ring=CyclotomicField(384))
M = H._module
chi = DirichletCharacter(H, M([192,255,64]))
chi.galois_orbit()
[g,chi] = znchar(Mod(11,4608))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4608\) | |
| Conductor: | \(4608\) | 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: | even | 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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4608}(11,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{384}\right)\) | \(e\left(\frac{59}{192}\right)\) | \(e\left(\frac{43}{384}\right)\) | \(e\left(\frac{17}{384}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{99}{128}\right)\) | \(e\left(\frac{121}{192}\right)\) | \(e\left(\frac{191}{192}\right)\) | \(e\left(\frac{325}{384}\right)\) | \(e\left(\frac{7}{48}\right)\) |
| \(\chi_{4608}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{307}{384}\right)\) | \(e\left(\frac{127}{192}\right)\) | \(e\left(\frac{47}{384}\right)\) | \(e\left(\frac{349}{384}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{7}{128}\right)\) | \(e\left(\frac{101}{192}\right)\) | \(e\left(\frac{115}{192}\right)\) | \(e\left(\frac{257}{384}\right)\) | \(e\left(\frac{11}{48}\right)\) |
| \(\chi_{4608}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{245}{384}\right)\) | \(e\left(\frac{137}{192}\right)\) | \(e\left(\frac{25}{384}\right)\) | \(e\left(\frac{251}{384}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{1}{128}\right)\) | \(e\left(\frac{115}{192}\right)\) | \(e\left(\frac{53}{192}\right)\) | \(e\left(\frac{55}{384}\right)\) | \(e\left(\frac{13}{48}\right)\) |
| \(\chi_{4608}(131,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{265}{384}\right)\) | \(e\left(\frac{109}{192}\right)\) | \(e\left(\frac{317}{384}\right)\) | \(e\left(\frac{295}{384}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{69}{128}\right)\) | \(e\left(\frac{191}{192}\right)\) | \(e\left(\frac{73}{192}\right)\) | \(e\left(\frac{83}{384}\right)\) | \(e\left(\frac{17}{48}\right)\) |
| \(\chi_{4608}(155,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{347}{384}\right)\) | \(e\left(\frac{71}{192}\right)\) | \(e\left(\frac{247}{384}\right)\) | \(e\left(\frac{53}{384}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{15}{128}\right)\) | \(e\left(\frac{61}{192}\right)\) | \(e\left(\frac{155}{192}\right)\) | \(e\left(\frac{313}{384}\right)\) | \(e\left(\frac{19}{48}\right)\) |
| \(\chi_{4608}(203,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{271}{384}\right)\) | \(e\left(\frac{139}{192}\right)\) | \(e\left(\frac{59}{384}\right)\) | \(e\left(\frac{193}{384}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{115}{128}\right)\) | \(e\left(\frac{41}{192}\right)\) | \(e\left(\frac{79}{192}\right)\) | \(e\left(\frac{53}{384}\right)\) | \(e\left(\frac{23}{48}\right)\) |
| \(\chi_{4608}(227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{384}\right)\) | \(e\left(\frac{53}{192}\right)\) | \(e\left(\frac{325}{384}\right)\) | \(e\left(\frac{191}{384}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{13}{128}\right)\) | \(e\left(\frac{151}{192}\right)\) | \(e\left(\frac{113}{192}\right)\) | \(e\left(\frac{331}{384}\right)\) | \(e\left(\frac{25}{48}\right)\) |
| \(\chi_{4608}(275,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{325}{384}\right)\) | \(e\left(\frac{25}{192}\right)\) | \(e\left(\frac{41}{384}\right)\) | \(e\left(\frac{43}{384}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{17}{128}\right)\) | \(e\left(\frac{35}{192}\right)\) | \(e\left(\frac{133}{192}\right)\) | \(e\left(\frac{167}{384}\right)\) | \(e\left(\frac{29}{48}\right)\) |
| \(\chi_{4608}(299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{311}{384}\right)\) | \(e\left(\frac{83}{192}\right)\) | \(e\left(\frac{259}{384}\right)\) | \(e\left(\frac{281}{384}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{123}{128}\right)\) | \(e\left(\frac{1}{192}\right)\) | \(e\left(\frac{119}{192}\right)\) | \(e\left(\frac{109}{384}\right)\) | \(e\left(\frac{31}{48}\right)\) |
| \(\chi_{4608}(347,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{384}\right)\) | \(e\left(\frac{151}{192}\right)\) | \(e\left(\frac{263}{384}\right)\) | \(e\left(\frac{229}{384}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{31}{128}\right)\) | \(e\left(\frac{173}{192}\right)\) | \(e\left(\frac{43}{192}\right)\) | \(e\left(\frac{41}{384}\right)\) | \(e\left(\frac{35}{48}\right)\) |
| \(\chi_{4608}(371,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{384}\right)\) | \(e\left(\frac{161}{192}\right)\) | \(e\left(\frac{49}{384}\right)\) | \(e\left(\frac{323}{384}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{89}{128}\right)\) | \(e\left(\frac{187}{192}\right)\) | \(e\left(\frac{173}{192}\right)\) | \(e\left(\frac{31}{384}\right)\) | \(e\left(\frac{37}{48}\right)\) |
| \(\chi_{4608}(419,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{384}\right)\) | \(e\left(\frac{133}{192}\right)\) | \(e\left(\frac{341}{384}\right)\) | \(e\left(\frac{367}{384}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{29}{128}\right)\) | \(e\left(\frac{71}{192}\right)\) | \(e\left(\frac{1}{192}\right)\) | \(e\left(\frac{59}{384}\right)\) | \(e\left(\frac{41}{48}\right)\) |
| \(\chi_{4608}(443,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{384}\right)\) | \(e\left(\frac{95}{192}\right)\) | \(e\left(\frac{79}{384}\right)\) | \(e\left(\frac{317}{384}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{39}{128}\right)\) | \(e\left(\frac{133}{192}\right)\) | \(e\left(\frac{83}{192}\right)\) | \(e\left(\frac{97}{384}\right)\) | \(e\left(\frac{43}{48}\right)\) |
| \(\chi_{4608}(491,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{384}\right)\) | \(e\left(\frac{163}{192}\right)\) | \(e\left(\frac{275}{384}\right)\) | \(e\left(\frac{73}{384}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{11}{128}\right)\) | \(e\left(\frac{113}{192}\right)\) | \(e\left(\frac{7}{192}\right)\) | \(e\left(\frac{221}{384}\right)\) | \(e\left(\frac{47}{48}\right)\) |
| \(\chi_{4608}(515,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{384}\right)\) | \(e\left(\frac{77}{192}\right)\) | \(e\left(\frac{349}{384}\right)\) | \(e\left(\frac{263}{384}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{101}{128}\right)\) | \(e\left(\frac{31}{192}\right)\) | \(e\left(\frac{41}{192}\right)\) | \(e\left(\frac{307}{384}\right)\) | \(e\left(\frac{1}{48}\right)\) |
| \(\chi_{4608}(563,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{253}{384}\right)\) | \(e\left(\frac{49}{192}\right)\) | \(e\left(\frac{65}{384}\right)\) | \(e\left(\frac{115}{384}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{105}{128}\right)\) | \(e\left(\frac{107}{192}\right)\) | \(e\left(\frac{61}{192}\right)\) | \(e\left(\frac{143}{384}\right)\) | \(e\left(\frac{5}{48}\right)\) |
| \(\chi_{4608}(587,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{384}\right)\) | \(e\left(\frac{107}{192}\right)\) | \(e\left(\frac{91}{384}\right)\) | \(e\left(\frac{161}{384}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{19}{128}\right)\) | \(e\left(\frac{73}{192}\right)\) | \(e\left(\frac{47}{192}\right)\) | \(e\left(\frac{277}{384}\right)\) | \(e\left(\frac{7}{48}\right)\) |
| \(\chi_{4608}(635,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{384}\right)\) | \(e\left(\frac{175}{192}\right)\) | \(e\left(\frac{95}{384}\right)\) | \(e\left(\frac{109}{384}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{55}{128}\right)\) | \(e\left(\frac{53}{192}\right)\) | \(e\left(\frac{163}{192}\right)\) | \(e\left(\frac{209}{384}\right)\) | \(e\left(\frac{11}{48}\right)\) |
| \(\chi_{4608}(659,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{384}\right)\) | \(e\left(\frac{185}{192}\right)\) | \(e\left(\frac{73}{384}\right)\) | \(e\left(\frac{11}{384}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{49}{128}\right)\) | \(e\left(\frac{67}{192}\right)\) | \(e\left(\frac{101}{192}\right)\) | \(e\left(\frac{7}{384}\right)\) | \(e\left(\frac{13}{48}\right)\) |
| \(\chi_{4608}(707,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{384}\right)\) | \(e\left(\frac{157}{192}\right)\) | \(e\left(\frac{365}{384}\right)\) | \(e\left(\frac{55}{384}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{117}{128}\right)\) | \(e\left(\frac{143}{192}\right)\) | \(e\left(\frac{121}{192}\right)\) | \(e\left(\frac{35}{384}\right)\) | \(e\left(\frac{17}{48}\right)\) |
| \(\chi_{4608}(731,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{203}{384}\right)\) | \(e\left(\frac{119}{192}\right)\) | \(e\left(\frac{295}{384}\right)\) | \(e\left(\frac{197}{384}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{63}{128}\right)\) | \(e\left(\frac{13}{192}\right)\) | \(e\left(\frac{11}{192}\right)\) | \(e\left(\frac{265}{384}\right)\) | \(e\left(\frac{19}{48}\right)\) |
| \(\chi_{4608}(779,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{384}\right)\) | \(e\left(\frac{187}{192}\right)\) | \(e\left(\frac{107}{384}\right)\) | \(e\left(\frac{337}{384}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{35}{128}\right)\) | \(e\left(\frac{185}{192}\right)\) | \(e\left(\frac{127}{192}\right)\) | \(e\left(\frac{5}{384}\right)\) | \(e\left(\frac{23}{48}\right)\) |
| \(\chi_{4608}(803,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{353}{384}\right)\) | \(e\left(\frac{101}{192}\right)\) | \(e\left(\frac{373}{384}\right)\) | \(e\left(\frac{335}{384}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{61}{128}\right)\) | \(e\left(\frac{103}{192}\right)\) | \(e\left(\frac{161}{192}\right)\) | \(e\left(\frac{283}{384}\right)\) | \(e\left(\frac{25}{48}\right)\) |
| \(\chi_{4608}(851,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{181}{384}\right)\) | \(e\left(\frac{73}{192}\right)\) | \(e\left(\frac{89}{384}\right)\) | \(e\left(\frac{187}{384}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{65}{128}\right)\) | \(e\left(\frac{179}{192}\right)\) | \(e\left(\frac{181}{192}\right)\) | \(e\left(\frac{119}{384}\right)\) | \(e\left(\frac{29}{48}\right)\) |
| \(\chi_{4608}(875,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{384}\right)\) | \(e\left(\frac{131}{192}\right)\) | \(e\left(\frac{307}{384}\right)\) | \(e\left(\frac{41}{384}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{43}{128}\right)\) | \(e\left(\frac{145}{192}\right)\) | \(e\left(\frac{167}{192}\right)\) | \(e\left(\frac{61}{384}\right)\) | \(e\left(\frac{31}{48}\right)\) |
| \(\chi_{4608}(923,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{283}{384}\right)\) | \(e\left(\frac{7}{192}\right)\) | \(e\left(\frac{311}{384}\right)\) | \(e\left(\frac{373}{384}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{79}{128}\right)\) | \(e\left(\frac{125}{192}\right)\) | \(e\left(\frac{91}{192}\right)\) | \(e\left(\frac{377}{384}\right)\) | \(e\left(\frac{35}{48}\right)\) |
| \(\chi_{4608}(947,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{384}\right)\) | \(e\left(\frac{17}{192}\right)\) | \(e\left(\frac{97}{384}\right)\) | \(e\left(\frac{83}{384}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{9}{128}\right)\) | \(e\left(\frac{139}{192}\right)\) | \(e\left(\frac{29}{192}\right)\) | \(e\left(\frac{367}{384}\right)\) | \(e\left(\frac{37}{48}\right)\) |
| \(\chi_{4608}(995,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{384}\right)\) | \(e\left(\frac{181}{192}\right)\) | \(e\left(\frac{5}{384}\right)\) | \(e\left(\frac{127}{384}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{77}{128}\right)\) | \(e\left(\frac{23}{192}\right)\) | \(e\left(\frac{49}{192}\right)\) | \(e\left(\frac{11}{384}\right)\) | \(e\left(\frac{41}{48}\right)\) |
| \(\chi_{4608}(1019,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{323}{384}\right)\) | \(e\left(\frac{143}{192}\right)\) | \(e\left(\frac{127}{384}\right)\) | \(e\left(\frac{77}{384}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{87}{128}\right)\) | \(e\left(\frac{85}{192}\right)\) | \(e\left(\frac{131}{192}\right)\) | \(e\left(\frac{49}{384}\right)\) | \(e\left(\frac{43}{48}\right)\) |
| \(\chi_{4608}(1067,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{247}{384}\right)\) | \(e\left(\frac{19}{192}\right)\) | \(e\left(\frac{323}{384}\right)\) | \(e\left(\frac{217}{384}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{59}{128}\right)\) | \(e\left(\frac{65}{192}\right)\) | \(e\left(\frac{55}{192}\right)\) | \(e\left(\frac{173}{384}\right)\) | \(e\left(\frac{47}{48}\right)\) |
| \(\chi_{4608}(1091,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{281}{384}\right)\) | \(e\left(\frac{125}{192}\right)\) | \(e\left(\frac{13}{384}\right)\) | \(e\left(\frac{23}{384}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{21}{128}\right)\) | \(e\left(\frac{175}{192}\right)\) | \(e\left(\frac{89}{192}\right)\) | \(e\left(\frac{259}{384}\right)\) | \(e\left(\frac{1}{48}\right)\) |