from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3072, base_ring=CyclotomicField(256))
M = H._module
chi = DirichletCharacter(H, M([0,239,0]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,3072))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3072\) | |
| Conductor: | \(1024\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(256\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1024.q | 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: | $\Q(\zeta_{256})$ |
| Fixed field: | Number field defined by a degree 256 polynomial (not computed) |
First 31 of 128 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3072}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{239}{256}\right)\) | \(e\left(\frac{75}{128}\right)\) | \(e\left(\frac{219}{256}\right)\) | \(e\left(\frac{33}{256}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{249}{256}\right)\) | \(e\left(\frac{73}{128}\right)\) | \(e\left(\frac{111}{128}\right)\) | \(e\left(\frac{213}{256}\right)\) | \(e\left(\frac{31}{32}\right)\) |
| \(\chi_{3072}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{256}\right)\) | \(e\left(\frac{93}{128}\right)\) | \(e\left(\frac{205}{256}\right)\) | \(e\left(\frac{87}{256}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{191}{256}\right)\) | \(e\left(\frac{111}{128}\right)\) | \(e\left(\frac{25}{128}\right)\) | \(e\left(\frac{3}{256}\right)\) | \(e\left(\frac{9}{32}\right)\) |
| \(\chi_{3072}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{256}\right)\) | \(e\left(\frac{63}{128}\right)\) | \(e\left(\frac{15}{256}\right)\) | \(e\left(\frac{125}{256}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{245}{256}\right)\) | \(e\left(\frac{5}{128}\right)\) | \(e\left(\frac{83}{128}\right)\) | \(e\left(\frac{225}{256}\right)\) | \(e\left(\frac{3}{32}\right)\) |
| \(\chi_{3072}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{157}{256}\right)\) | \(e\left(\frac{113}{128}\right)\) | \(e\left(\frac{161}{256}\right)\) | \(e\left(\frac{147}{256}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{155}{256}\right)\) | \(e\left(\frac{11}{128}\right)\) | \(e\left(\frac{29}{128}\right)\) | \(e\left(\frac{111}{256}\right)\) | \(e\left(\frac{13}{32}\right)\) |
| \(\chi_{3072}(109,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{247}{256}\right)\) | \(e\left(\frac{115}{128}\right)\) | \(e\left(\frac{131}{256}\right)\) | \(e\left(\frac{153}{256}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{177}{256}\right)\) | \(e\left(\frac{1}{128}\right)\) | \(e\left(\frac{119}{128}\right)\) | \(e\left(\frac{173}{256}\right)\) | \(e\left(\frac{7}{32}\right)\) |
| \(\chi_{3072}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{256}\right)\) | \(e\left(\frac{69}{128}\right)\) | \(e\left(\frac{181}{256}\right)\) | \(e\left(\frac{143}{256}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{55}{256}\right)\) | \(e\left(\frac{103}{128}\right)\) | \(e\left(\frac{97}{128}\right)\) | \(e\left(\frac{155}{256}\right)\) | \(e\left(\frac{17}{32}\right)\) |
| \(\chi_{3072}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{219}{256}\right)\) | \(e\left(\frac{103}{128}\right)\) | \(e\left(\frac{55}{256}\right)\) | \(e\left(\frac{117}{256}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{45}{256}\right)\) | \(e\left(\frac{61}{128}\right)\) | \(e\left(\frac{91}{128}\right)\) | \(e\left(\frac{57}{256}\right)\) | \(e\left(\frac{11}{32}\right)\) |
| \(\chi_{3072}(181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{256}\right)\) | \(e\left(\frac{89}{128}\right)\) | \(e\left(\frac{9}{256}\right)\) | \(e\left(\frac{75}{256}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{147}{256}\right)\) | \(e\left(\frac{3}{128}\right)\) | \(e\left(\frac{101}{128}\right)\) | \(e\left(\frac{135}{256}\right)\) | \(e\left(\frac{21}{32}\right)\) |
| \(\chi_{3072}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{255}{256}\right)\) | \(e\left(\frac{27}{128}\right)\) | \(e\left(\frac{43}{256}\right)\) | \(e\left(\frac{17}{256}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{105}{256}\right)\) | \(e\left(\frac{57}{128}\right)\) | \(e\left(\frac{127}{128}\right)\) | \(e\left(\frac{133}{256}\right)\) | \(e\left(\frac{15}{32}\right)\) |
| \(\chi_{3072}(229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{169}{256}\right)\) | \(e\left(\frac{45}{128}\right)\) | \(e\left(\frac{157}{256}\right)\) | \(e\left(\frac{199}{256}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{175}{256}\right)\) | \(e\left(\frac{95}{128}\right)\) | \(e\left(\frac{41}{128}\right)\) | \(e\left(\frac{51}{256}\right)\) | \(e\left(\frac{25}{32}\right)\) |
| \(\chi_{3072}(253,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{99}{256}\right)\) | \(e\left(\frac{15}{128}\right)\) | \(e\left(\frac{95}{256}\right)\) | \(e\left(\frac{109}{256}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{101}{256}\right)\) | \(e\left(\frac{117}{128}\right)\) | \(e\left(\frac{99}{128}\right)\) | \(e\left(\frac{145}{256}\right)\) | \(e\left(\frac{19}{32}\right)\) |
| \(\chi_{3072}(277,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{256}\right)\) | \(e\left(\frac{65}{128}\right)\) | \(e\left(\frac{113}{256}\right)\) | \(e\left(\frac{3}{256}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{139}{256}\right)\) | \(e\left(\frac{123}{128}\right)\) | \(e\left(\frac{45}{128}\right)\) | \(e\left(\frac{159}{256}\right)\) | \(e\left(\frac{29}{32}\right)\) |
| \(\chi_{3072}(301,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{256}\right)\) | \(e\left(\frac{67}{128}\right)\) | \(e\left(\frac{211}{256}\right)\) | \(e\left(\frac{137}{256}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{33}{256}\right)\) | \(e\left(\frac{113}{128}\right)\) | \(e\left(\frac{7}{128}\right)\) | \(e\left(\frac{93}{256}\right)\) | \(e\left(\frac{23}{32}\right)\) |
| \(\chi_{3072}(325,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{241}{256}\right)\) | \(e\left(\frac{21}{128}\right)\) | \(e\left(\frac{133}{256}\right)\) | \(e\left(\frac{255}{256}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{39}{256}\right)\) | \(e\left(\frac{87}{128}\right)\) | \(e\left(\frac{113}{128}\right)\) | \(e\left(\frac{203}{256}\right)\) | \(e\left(\frac{1}{32}\right)\) |
| \(\chi_{3072}(349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{235}{256}\right)\) | \(e\left(\frac{55}{128}\right)\) | \(e\left(\frac{135}{256}\right)\) | \(e\left(\frac{101}{256}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{157}{256}\right)\) | \(e\left(\frac{45}{128}\right)\) | \(e\left(\frac{107}{128}\right)\) | \(e\left(\frac{233}{256}\right)\) | \(e\left(\frac{27}{32}\right)\) |
| \(\chi_{3072}(373,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{245}{256}\right)\) | \(e\left(\frac{41}{128}\right)\) | \(e\left(\frac{217}{256}\right)\) | \(e\left(\frac{187}{256}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{131}{256}\right)\) | \(e\left(\frac{115}{128}\right)\) | \(e\left(\frac{117}{128}\right)\) | \(e\left(\frac{183}{256}\right)\) | \(e\left(\frac{5}{32}\right)\) |
| \(\chi_{3072}(397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{256}\right)\) | \(e\left(\frac{107}{128}\right)\) | \(e\left(\frac{123}{256}\right)\) | \(e\left(\frac{1}{256}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{217}{256}\right)\) | \(e\left(\frac{41}{128}\right)\) | \(e\left(\frac{15}{128}\right)\) | \(e\left(\frac{53}{256}\right)\) | \(e\left(\frac{31}{32}\right)\) |
| \(\chi_{3072}(421,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{256}\right)\) | \(e\left(\frac{125}{128}\right)\) | \(e\left(\frac{109}{256}\right)\) | \(e\left(\frac{55}{256}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{159}{256}\right)\) | \(e\left(\frac{79}{128}\right)\) | \(e\left(\frac{57}{128}\right)\) | \(e\left(\frac{99}{256}\right)\) | \(e\left(\frac{9}{32}\right)\) |
| \(\chi_{3072}(445,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{256}\right)\) | \(e\left(\frac{95}{128}\right)\) | \(e\left(\frac{175}{256}\right)\) | \(e\left(\frac{93}{256}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{213}{256}\right)\) | \(e\left(\frac{101}{128}\right)\) | \(e\left(\frac{115}{128}\right)\) | \(e\left(\frac{65}{256}\right)\) | \(e\left(\frac{3}{32}\right)\) |
| \(\chi_{3072}(469,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{189}{256}\right)\) | \(e\left(\frac{17}{128}\right)\) | \(e\left(\frac{65}{256}\right)\) | \(e\left(\frac{115}{256}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{123}{256}\right)\) | \(e\left(\frac{107}{128}\right)\) | \(e\left(\frac{61}{128}\right)\) | \(e\left(\frac{207}{256}\right)\) | \(e\left(\frac{13}{32}\right)\) |
| \(\chi_{3072}(493,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{256}\right)\) | \(e\left(\frac{19}{128}\right)\) | \(e\left(\frac{35}{256}\right)\) | \(e\left(\frac{121}{256}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{145}{256}\right)\) | \(e\left(\frac{97}{128}\right)\) | \(e\left(\frac{23}{128}\right)\) | \(e\left(\frac{13}{256}\right)\) | \(e\left(\frac{7}{32}\right)\) |
| \(\chi_{3072}(517,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{129}{256}\right)\) | \(e\left(\frac{101}{128}\right)\) | \(e\left(\frac{85}{256}\right)\) | \(e\left(\frac{111}{256}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{23}{256}\right)\) | \(e\left(\frac{71}{128}\right)\) | \(e\left(\frac{1}{128}\right)\) | \(e\left(\frac{251}{256}\right)\) | \(e\left(\frac{17}{32}\right)\) |
| \(\chi_{3072}(541,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{256}\right)\) | \(e\left(\frac{7}{128}\right)\) | \(e\left(\frac{215}{256}\right)\) | \(e\left(\frac{85}{256}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{13}{256}\right)\) | \(e\left(\frac{29}{128}\right)\) | \(e\left(\frac{123}{128}\right)\) | \(e\left(\frac{153}{256}\right)\) | \(e\left(\frac{11}{32}\right)\) |
| \(\chi_{3072}(565,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{256}\right)\) | \(e\left(\frac{121}{128}\right)\) | \(e\left(\frac{169}{256}\right)\) | \(e\left(\frac{43}{256}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{115}{256}\right)\) | \(e\left(\frac{99}{128}\right)\) | \(e\left(\frac{5}{128}\right)\) | \(e\left(\frac{231}{256}\right)\) | \(e\left(\frac{21}{32}\right)\) |
| \(\chi_{3072}(589,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{256}\right)\) | \(e\left(\frac{59}{128}\right)\) | \(e\left(\frac{203}{256}\right)\) | \(e\left(\frac{241}{256}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{73}{256}\right)\) | \(e\left(\frac{25}{128}\right)\) | \(e\left(\frac{31}{128}\right)\) | \(e\left(\frac{229}{256}\right)\) | \(e\left(\frac{15}{32}\right)\) |
| \(\chi_{3072}(613,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{201}{256}\right)\) | \(e\left(\frac{77}{128}\right)\) | \(e\left(\frac{61}{256}\right)\) | \(e\left(\frac{167}{256}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{143}{256}\right)\) | \(e\left(\frac{63}{128}\right)\) | \(e\left(\frac{73}{128}\right)\) | \(e\left(\frac{147}{256}\right)\) | \(e\left(\frac{25}{32}\right)\) |
| \(\chi_{3072}(637,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{256}\right)\) | \(e\left(\frac{47}{128}\right)\) | \(e\left(\frac{255}{256}\right)\) | \(e\left(\frac{77}{256}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{69}{256}\right)\) | \(e\left(\frac{85}{128}\right)\) | \(e\left(\frac{3}{128}\right)\) | \(e\left(\frac{241}{256}\right)\) | \(e\left(\frac{19}{32}\right)\) |
| \(\chi_{3072}(661,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{256}\right)\) | \(e\left(\frac{97}{128}\right)\) | \(e\left(\frac{17}{256}\right)\) | \(e\left(\frac{227}{256}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{107}{256}\right)\) | \(e\left(\frac{91}{128}\right)\) | \(e\left(\frac{77}{128}\right)\) | \(e\left(\frac{255}{256}\right)\) | \(e\left(\frac{29}{32}\right)\) |
| \(\chi_{3072}(685,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{256}\right)\) | \(e\left(\frac{99}{128}\right)\) | \(e\left(\frac{115}{256}\right)\) | \(e\left(\frac{105}{256}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{1}{256}\right)\) | \(e\left(\frac{81}{128}\right)\) | \(e\left(\frac{39}{128}\right)\) | \(e\left(\frac{189}{256}\right)\) | \(e\left(\frac{23}{32}\right)\) |
| \(\chi_{3072}(709,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{256}\right)\) | \(e\left(\frac{53}{128}\right)\) | \(e\left(\frac{37}{256}\right)\) | \(e\left(\frac{223}{256}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{7}{256}\right)\) | \(e\left(\frac{55}{128}\right)\) | \(e\left(\frac{17}{128}\right)\) | \(e\left(\frac{43}{256}\right)\) | \(e\left(\frac{1}{32}\right)\) |
| \(\chi_{3072}(733,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{256}\right)\) | \(e\left(\frac{87}{128}\right)\) | \(e\left(\frac{39}{256}\right)\) | \(e\left(\frac{69}{256}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{125}{256}\right)\) | \(e\left(\frac{13}{128}\right)\) | \(e\left(\frac{11}{128}\right)\) | \(e\left(\frac{73}{256}\right)\) | \(e\left(\frac{27}{32}\right)\) |