from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3072, base_ring=CyclotomicField(128))
M = H._module
chi = DirichletCharacter(H, M([64,101,0]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,3072))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3072\) | |
| Conductor: | \(512\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(128\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 512.p | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | 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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3072}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{128}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{73}{128}\right)\) | \(e\left(\frac{75}{128}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{83}{128}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{7}{128}\right)\) | \(e\left(\frac{13}{16}\right)\) |
| \(\chi_{3072}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{128}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{71}{128}\right)\) | \(e\left(\frac{101}{128}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{93}{128}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{105}{128}\right)\) | \(e\left(\frac{3}{16}\right)\) |
| \(\chi_{3072}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{128}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{101}{128}\right)\) | \(e\left(\frac{95}{128}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{71}{128}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{43}{128}\right)\) | \(e\left(\frac{9}{16}\right)\) |
| \(\chi_{3072}(151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{87}{128}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{35}{128}\right)\) | \(e\left(\frac{57}{128}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{17}{128}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{77}{128}\right)\) | \(e\left(\frac{15}{16}\right)\) |
| \(\chi_{3072}(199,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{128}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{1}{128}\right)\) | \(e\left(\frac{115}{128}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{59}{128}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{79}{128}\right)\) | \(e\left(\frac{5}{16}\right)\) |
| \(\chi_{3072}(247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{128}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{127}{128}\right)\) | \(e\left(\frac{13}{128}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{69}{128}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{49}{128}\right)\) | \(e\left(\frac{11}{16}\right)\) |
| \(\chi_{3072}(295,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{105}{128}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{29}{128}\right)\) | \(e\left(\frac{7}{128}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{47}{128}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{115}{128}\right)\) | \(e\left(\frac{1}{16}\right)\) |
| \(\chi_{3072}(343,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{128}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{91}{128}\right)\) | \(e\left(\frac{97}{128}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{121}{128}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{21}{128}\right)\) | \(e\left(\frac{7}{16}\right)\) |
| \(\chi_{3072}(391,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{128}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{57}{128}\right)\) | \(e\left(\frac{27}{128}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{35}{128}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{23}{128}\right)\) | \(e\left(\frac{13}{16}\right)\) |
| \(\chi_{3072}(439,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{128}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{55}{128}\right)\) | \(e\left(\frac{53}{128}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{45}{128}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{121}{128}\right)\) | \(e\left(\frac{3}{16}\right)\) |
| \(\chi_{3072}(487,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{128}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{85}{128}\right)\) | \(e\left(\frac{47}{128}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{23}{128}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{59}{128}\right)\) | \(e\left(\frac{9}{16}\right)\) |
| \(\chi_{3072}(535,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{128}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{19}{128}\right)\) | \(e\left(\frac{9}{128}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{97}{128}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{93}{128}\right)\) | \(e\left(\frac{15}{16}\right)\) |
| \(\chi_{3072}(583,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{128}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{113}{128}\right)\) | \(e\left(\frac{67}{128}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{11}{128}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{95}{128}\right)\) | \(e\left(\frac{5}{16}\right)\) |
| \(\chi_{3072}(631,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{128}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{111}{128}\right)\) | \(e\left(\frac{93}{128}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{21}{128}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{65}{128}\right)\) | \(e\left(\frac{11}{16}\right)\) |
| \(\chi_{3072}(679,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{128}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{13}{128}\right)\) | \(e\left(\frac{87}{128}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{127}{128}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{3}{128}\right)\) | \(e\left(\frac{1}{16}\right)\) |
| \(\chi_{3072}(727,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{128}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{75}{128}\right)\) | \(e\left(\frac{49}{128}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{73}{128}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{37}{128}\right)\) | \(e\left(\frac{7}{16}\right)\) |
| \(\chi_{3072}(775,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{128}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{41}{128}\right)\) | \(e\left(\frac{107}{128}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{115}{128}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{39}{128}\right)\) | \(e\left(\frac{13}{16}\right)\) |
| \(\chi_{3072}(823,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{75}{128}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{39}{128}\right)\) | \(e\left(\frac{5}{128}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{125}{128}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{9}{128}\right)\) | \(e\left(\frac{3}{16}\right)\) |
| \(\chi_{3072}(871,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{128}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{69}{128}\right)\) | \(e\left(\frac{127}{128}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{103}{128}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{75}{128}\right)\) | \(e\left(\frac{9}{16}\right)\) |
| \(\chi_{3072}(919,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{128}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{3}{128}\right)\) | \(e\left(\frac{89}{128}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{49}{128}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{109}{128}\right)\) | \(e\left(\frac{15}{16}\right)\) |
| \(\chi_{3072}(967,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{128}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{97}{128}\right)\) | \(e\left(\frac{19}{128}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{91}{128}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{111}{128}\right)\) | \(e\left(\frac{5}{16}\right)\) |
| \(\chi_{3072}(1015,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{128}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{95}{128}\right)\) | \(e\left(\frac{45}{128}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{101}{128}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{81}{128}\right)\) | \(e\left(\frac{11}{16}\right)\) |
| \(\chi_{3072}(1063,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{128}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{125}{128}\right)\) | \(e\left(\frac{39}{128}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{79}{128}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{19}{128}\right)\) | \(e\left(\frac{1}{16}\right)\) |
| \(\chi_{3072}(1111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{128}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{59}{128}\right)\) | \(e\left(\frac{1}{128}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{25}{128}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{53}{128}\right)\) | \(e\left(\frac{7}{16}\right)\) |
| \(\chi_{3072}(1159,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{117}{128}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{25}{128}\right)\) | \(e\left(\frac{59}{128}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{67}{128}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{55}{128}\right)\) | \(e\left(\frac{13}{16}\right)\) |
| \(\chi_{3072}(1207,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{123}{128}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{23}{128}\right)\) | \(e\left(\frac{85}{128}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{77}{128}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{25}{128}\right)\) | \(e\left(\frac{3}{16}\right)\) |
| \(\chi_{3072}(1255,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{128}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{53}{128}\right)\) | \(e\left(\frac{79}{128}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{55}{128}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{91}{128}\right)\) | \(e\left(\frac{9}{16}\right)\) |
| \(\chi_{3072}(1303,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{128}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{115}{128}\right)\) | \(e\left(\frac{41}{128}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{1}{128}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{125}{128}\right)\) | \(e\left(\frac{15}{16}\right)\) |
| \(\chi_{3072}(1351,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{128}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{81}{128}\right)\) | \(e\left(\frac{99}{128}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{43}{128}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{127}{128}\right)\) | \(e\left(\frac{5}{16}\right)\) |
| \(\chi_{3072}(1399,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{128}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{79}{128}\right)\) | \(e\left(\frac{125}{128}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{53}{128}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{97}{128}\right)\) | \(e\left(\frac{11}{16}\right)\) |
| \(\chi_{3072}(1447,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{128}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{109}{128}\right)\) | \(e\left(\frac{119}{128}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{31}{128}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{35}{128}\right)\) | \(e\left(\frac{1}{16}\right)\) |