from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3584, base_ring=CyclotomicField(192))
M = H._module
chi = DirichletCharacter(H, M([0,81,32]))
chi.galois_orbit()
[g,chi] = znchar(Mod(73,3584))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3584\) | |
| Conductor: | \(1792\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(192\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1792.ca | 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: | $\Q(\zeta_{192})$ |
| Fixed field: | Number field defined by a degree 192 polynomial (not computed) |
First 31 of 64 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3584}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{192}\right)\) | \(e\left(\frac{49}{192}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{101}{192}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{103}{192}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{49}{96}\right)\) |
| \(\chi_{3584}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{192}\right)\) | \(e\left(\frac{11}{192}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{7}{192}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{125}{192}\right)\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{11}{96}\right)\) |
| \(\chi_{3584}(185,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{192}\right)\) | \(e\left(\frac{7}{192}\right)\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{179}{192}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{97}{192}\right)\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{7}{96}\right)\) |
| \(\chi_{3584}(201,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{192}\right)\) | \(e\left(\frac{65}{192}\right)\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{181}{192}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{23}{192}\right)\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{65}{96}\right)\) |
| \(\chi_{3584}(297,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{192}\right)\) | \(e\left(\frac{61}{192}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{161}{192}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{187}{192}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{61}{96}\right)\) |
| \(\chi_{3584}(313,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{192}\right)\) | \(e\left(\frac{23}{192}\right)\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{67}{192}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{17}{192}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{23}{96}\right)\) |
| \(\chi_{3584}(409,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{192}\right)\) | \(e\left(\frac{19}{192}\right)\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{47}{192}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{181}{192}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{19}{96}\right)\) |
| \(\chi_{3584}(425,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{192}\right)\) | \(e\left(\frac{77}{192}\right)\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{49}{192}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{107}{192}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{77}{96}\right)\) |
| \(\chi_{3584}(521,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{192}\right)\) | \(e\left(\frac{73}{192}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{29}{192}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{79}{192}\right)\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{73}{96}\right)\) |
| \(\chi_{3584}(537,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{192}\right)\) | \(e\left(\frac{35}{192}\right)\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{127}{192}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{101}{192}\right)\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{35}{96}\right)\) |
| \(\chi_{3584}(633,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{192}\right)\) | \(e\left(\frac{31}{192}\right)\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{107}{192}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{73}{192}\right)\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{31}{96}\right)\) |
| \(\chi_{3584}(649,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{192}\right)\) | \(e\left(\frac{89}{192}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{109}{192}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{191}{192}\right)\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{89}{96}\right)\) |
| \(\chi_{3584}(745,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{192}\right)\) | \(e\left(\frac{85}{192}\right)\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{89}{192}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{163}{192}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{85}{96}\right)\) |
| \(\chi_{3584}(761,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{192}\right)\) | \(e\left(\frac{47}{192}\right)\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{187}{192}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{185}{192}\right)\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{47}{96}\right)\) |
| \(\chi_{3584}(857,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{192}\right)\) | \(e\left(\frac{43}{192}\right)\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{167}{192}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{157}{192}\right)\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{43}{96}\right)\) |
| \(\chi_{3584}(873,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{192}\right)\) | \(e\left(\frac{101}{192}\right)\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{169}{192}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{83}{192}\right)\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{5}{96}\right)\) |
| \(\chi_{3584}(969,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{192}\right)\) | \(e\left(\frac{97}{192}\right)\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{149}{192}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{55}{192}\right)\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{1}{96}\right)\) |
| \(\chi_{3584}(985,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{192}\right)\) | \(e\left(\frac{59}{192}\right)\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{55}{192}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{77}{192}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{59}{96}\right)\) |
| \(\chi_{3584}(1081,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{192}\right)\) | \(e\left(\frac{55}{192}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{35}{192}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{49}{192}\right)\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{55}{96}\right)\) |
| \(\chi_{3584}(1097,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{192}\right)\) | \(e\left(\frac{113}{192}\right)\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{37}{192}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{167}{192}\right)\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{17}{96}\right)\) |
| \(\chi_{3584}(1193,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{192}\right)\) | \(e\left(\frac{109}{192}\right)\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{17}{192}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{139}{192}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{13}{96}\right)\) |
| \(\chi_{3584}(1209,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{192}\right)\) | \(e\left(\frac{71}{192}\right)\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{115}{192}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{161}{192}\right)\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{71}{96}\right)\) |
| \(\chi_{3584}(1305,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{192}\right)\) | \(e\left(\frac{67}{192}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{95}{192}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{133}{192}\right)\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{67}{96}\right)\) |
| \(\chi_{3584}(1321,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{192}\right)\) | \(e\left(\frac{125}{192}\right)\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{97}{192}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{59}{192}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{29}{96}\right)\) |
| \(\chi_{3584}(1417,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{192}\right)\) | \(e\left(\frac{121}{192}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{77}{192}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{31}{192}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{25}{96}\right)\) |
| \(\chi_{3584}(1433,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{192}\right)\) | \(e\left(\frac{83}{192}\right)\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{175}{192}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{53}{192}\right)\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{83}{96}\right)\) |
| \(\chi_{3584}(1529,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{192}\right)\) | \(e\left(\frac{79}{192}\right)\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{155}{192}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{25}{192}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{79}{96}\right)\) |
| \(\chi_{3584}(1545,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{187}{192}\right)\) | \(e\left(\frac{137}{192}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{157}{192}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{143}{192}\right)\) | \(e\left(\frac{31}{96}\right)\) | \(e\left(\frac{41}{96}\right)\) |
| \(\chi_{3584}(1641,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{192}\right)\) | \(e\left(\frac{133}{192}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{137}{192}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{115}{192}\right)\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{37}{96}\right)\) |
| \(\chi_{3584}(1657,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{192}\right)\) | \(e\left(\frac{95}{192}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{43}{192}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{137}{192}\right)\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{95}{96}\right)\) |
| \(\chi_{3584}(1753,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{192}\right)\) | \(e\left(\frac{91}{192}\right)\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{23}{192}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{109}{192}\right)\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{91}{96}\right)\) |