from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(512, base_ring=CyclotomicField(128))
M = H._module
chi = DirichletCharacter(H, M([0,1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,512))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(512\) | |
| Conductor: | \(512\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(128\) | 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: | $\Q(\zeta_{128})$ |
| Fixed field: | Number field defined by a degree 128 polynomial (not computed) |
First 31 of 64 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{512}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{128}\right)\) | \(e\left(\frac{1}{128}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{85}{128}\right)\) | \(e\left(\frac{111}{128}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{23}{128}\right)\) | \(e\left(\frac{109}{128}\right)\) |
| \(\chi_{512}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{128}\right)\) | \(e\left(\frac{111}{128}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{91}{128}\right)\) | \(e\left(\frac{33}{128}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{121}{128}\right)\) | \(e\left(\frac{67}{128}\right)\) |
| \(\chi_{512}(21,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{128}\right)\) | \(e\left(\frac{109}{128}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{49}{128}\right)\) | \(e\left(\frac{67}{128}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{75}{128}\right)\) | \(e\left(\frac{105}{128}\right)\) |
| \(\chi_{512}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{128}\right)\) | \(e\left(\frac{123}{128}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{87}{128}\right)\) | \(e\left(\frac{85}{128}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{13}{128}\right)\) | \(e\left(\frac{95}{128}\right)\) |
| \(\chi_{512}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{128}\right)\) | \(e\left(\frac{25}{128}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{77}{128}\right)\) | \(e\left(\frac{87}{128}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{63}{128}\right)\) | \(e\left(\frac{37}{128}\right)\) |
| \(\chi_{512}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{128}\right)\) | \(e\left(\frac{71}{128}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{19}{128}\right)\) | \(e\left(\frac{73}{128}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{97}{128}\right)\) | \(e\left(\frac{59}{128}\right)\) |
| \(\chi_{512}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{128}\right)\) | \(e\left(\frac{5}{128}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{41}{128}\right)\) | \(e\left(\frac{43}{128}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{115}{128}\right)\) | \(e\left(\frac{33}{128}\right)\) |
| \(\chi_{512}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{128}\right)\) | \(e\left(\frac{83}{128}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{15}{128}\right)\) | \(e\left(\frac{125}{128}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{117}{128}\right)\) | \(e\left(\frac{87}{128}\right)\) |
| \(\chi_{512}(69,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{128}\right)\) | \(e\left(\frac{49}{128}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{69}{128}\right)\) | \(e\left(\frac{63}{128}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{103}{128}\right)\) | \(e\left(\frac{93}{128}\right)\) |
| \(\chi_{512}(77,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{128}\right)\) | \(e\left(\frac{31}{128}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{75}{128}\right)\) | \(e\left(\frac{113}{128}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{73}{128}\right)\) | \(e\left(\frac{51}{128}\right)\) |
| \(\chi_{512}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{128}\right)\) | \(e\left(\frac{29}{128}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{33}{128}\right)\) | \(e\left(\frac{19}{128}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{27}{128}\right)\) | \(e\left(\frac{89}{128}\right)\) |
| \(\chi_{512}(93,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{128}\right)\) | \(e\left(\frac{43}{128}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{71}{128}\right)\) | \(e\left(\frac{37}{128}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{93}{128}\right)\) | \(e\left(\frac{79}{128}\right)\) |
| \(\chi_{512}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{123}{128}\right)\) | \(e\left(\frac{73}{128}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{61}{128}\right)\) | \(e\left(\frac{39}{128}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{15}{128}\right)\) | \(e\left(\frac{21}{128}\right)\) |
| \(\chi_{512}(109,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{128}\right)\) | \(e\left(\frac{119}{128}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{3}{128}\right)\) | \(e\left(\frac{25}{128}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{49}{128}\right)\) | \(e\left(\frac{43}{128}\right)\) |
| \(\chi_{512}(117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{128}\right)\) | \(e\left(\frac{53}{128}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{25}{128}\right)\) | \(e\left(\frac{123}{128}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{67}{128}\right)\) | \(e\left(\frac{17}{128}\right)\) |
| \(\chi_{512}(125,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{105}{128}\right)\) | \(e\left(\frac{3}{128}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{127}{128}\right)\) | \(e\left(\frac{77}{128}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{69}{128}\right)\) | \(e\left(\frac{71}{128}\right)\) |
| \(\chi_{512}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{128}\right)\) | \(e\left(\frac{97}{128}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{53}{128}\right)\) | \(e\left(\frac{15}{128}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{55}{128}\right)\) | \(e\left(\frac{77}{128}\right)\) |
| \(\chi_{512}(141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{128}\right)\) | \(e\left(\frac{79}{128}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{59}{128}\right)\) | \(e\left(\frac{65}{128}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{25}{128}\right)\) | \(e\left(\frac{35}{128}\right)\) |
| \(\chi_{512}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{128}\right)\) | \(e\left(\frac{77}{128}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{17}{128}\right)\) | \(e\left(\frac{99}{128}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{107}{128}\right)\) | \(e\left(\frac{73}{128}\right)\) |
| \(\chi_{512}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{128}\right)\) | \(e\left(\frac{91}{128}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{55}{128}\right)\) | \(e\left(\frac{117}{128}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{45}{128}\right)\) | \(e\left(\frac{63}{128}\right)\) |
| \(\chi_{512}(165,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{128}\right)\) | \(e\left(\frac{121}{128}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{45}{128}\right)\) | \(e\left(\frac{119}{128}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{95}{128}\right)\) | \(e\left(\frac{5}{128}\right)\) |
| \(\chi_{512}(173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{128}\right)\) | \(e\left(\frac{39}{128}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{115}{128}\right)\) | \(e\left(\frac{105}{128}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{1}{128}\right)\) | \(e\left(\frac{27}{128}\right)\) |
| \(\chi_{512}(181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{128}\right)\) | \(e\left(\frac{101}{128}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{9}{128}\right)\) | \(e\left(\frac{75}{128}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{19}{128}\right)\) | \(e\left(\frac{1}{128}\right)\) |
| \(\chi_{512}(189,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{128}\right)\) | \(e\left(\frac{51}{128}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{111}{128}\right)\) | \(e\left(\frac{29}{128}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{21}{128}\right)\) | \(e\left(\frac{55}{128}\right)\) |
| \(\chi_{512}(197,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{128}\right)\) | \(e\left(\frac{17}{128}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{37}{128}\right)\) | \(e\left(\frac{95}{128}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{7}{128}\right)\) | \(e\left(\frac{61}{128}\right)\) |
| \(\chi_{512}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{128}\right)\) | \(e\left(\frac{127}{128}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{43}{128}\right)\) | \(e\left(\frac{17}{128}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{105}{128}\right)\) | \(e\left(\frac{19}{128}\right)\) |
| \(\chi_{512}(213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{128}\right)\) | \(e\left(\frac{125}{128}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{1}{128}\right)\) | \(e\left(\frac{51}{128}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{59}{128}\right)\) | \(e\left(\frac{57}{128}\right)\) |
| \(\chi_{512}(221,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{128}\right)\) | \(e\left(\frac{11}{128}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{39}{128}\right)\) | \(e\left(\frac{69}{128}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{125}{128}\right)\) | \(e\left(\frac{47}{128}\right)\) |
| \(\chi_{512}(229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{128}\right)\) | \(e\left(\frac{41}{128}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{29}{128}\right)\) | \(e\left(\frac{71}{128}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{47}{128}\right)\) | \(e\left(\frac{117}{128}\right)\) |
| \(\chi_{512}(237,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{128}\right)\) | \(e\left(\frac{87}{128}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{99}{128}\right)\) | \(e\left(\frac{57}{128}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{81}{128}\right)\) | \(e\left(\frac{11}{128}\right)\) |
| \(\chi_{512}(245,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{128}\right)\) | \(e\left(\frac{21}{128}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{121}{128}\right)\) | \(e\left(\frac{27}{128}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{99}{128}\right)\) | \(e\left(\frac{113}{128}\right)\) |