from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4608, base_ring=CyclotomicField(192))
M = H._module
chi = DirichletCharacter(H, M([96,111,128]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,4608))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4608\) | |
| Conductor: | \(2304\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(192\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2304.cc | 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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4608}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{175}{192}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{59}{192}\right)\) | \(e\left(\frac{97}{192}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{149}{192}\right)\) | \(e\left(\frac{11}{24}\right)\) |
| \(\chi_{4608}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{192}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{79}{192}\right)\) | \(e\left(\frac{29}{192}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{1}{192}\right)\) | \(e\left(\frac{7}{24}\right)\) |
| \(\chi_{4608}(151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{192}\right)\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{137}{192}\right)\) | \(e\left(\frac{43}{192}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{167}{192}\right)\) | \(e\left(\frac{17}{24}\right)\) |
| \(\chi_{4608}(247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{192}\right)\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{157}{192}\right)\) | \(e\left(\frac{167}{192}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{19}{192}\right)\) | \(e\left(\frac{13}{24}\right)\) |
| \(\chi_{4608}(295,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{187}{192}\right)\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{119}{192}\right)\) | \(e\left(\frac{85}{192}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{89}{192}\right)\) | \(e\left(\frac{23}{24}\right)\) |
| \(\chi_{4608}(391,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{192}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{139}{192}\right)\) | \(e\left(\frac{17}{192}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{133}{192}\right)\) | \(e\left(\frac{19}{24}\right)\) |
| \(\chi_{4608}(439,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{192}\right)\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{5}{192}\right)\) | \(e\left(\frac{31}{192}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{107}{192}\right)\) | \(e\left(\frac{5}{24}\right)\) |
| \(\chi_{4608}(535,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{192}\right)\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{25}{192}\right)\) | \(e\left(\frac{155}{192}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{151}{192}\right)\) | \(e\left(\frac{1}{24}\right)\) |
| \(\chi_{4608}(583,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{192}\right)\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{179}{192}\right)\) | \(e\left(\frac{73}{192}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{29}{192}\right)\) | \(e\left(\frac{11}{24}\right)\) |
| \(\chi_{4608}(679,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{192}\right)\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{7}{192}\right)\) | \(e\left(\frac{5}{192}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{73}{192}\right)\) | \(e\left(\frac{7}{24}\right)\) |
| \(\chi_{4608}(727,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{192}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{65}{192}\right)\) | \(e\left(\frac{19}{192}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{47}{192}\right)\) | \(e\left(\frac{17}{24}\right)\) |
| \(\chi_{4608}(823,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{192}\right)\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{85}{192}\right)\) | \(e\left(\frac{143}{192}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{91}{192}\right)\) | \(e\left(\frac{13}{24}\right)\) |
| \(\chi_{4608}(871,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{192}\right)\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{47}{192}\right)\) | \(e\left(\frac{61}{192}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{161}{192}\right)\) | \(e\left(\frac{23}{24}\right)\) |
| \(\chi_{4608}(967,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{192}\right)\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{67}{192}\right)\) | \(e\left(\frac{185}{192}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{13}{192}\right)\) | \(e\left(\frac{19}{24}\right)\) |
| \(\chi_{4608}(1015,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{169}{192}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{125}{192}\right)\) | \(e\left(\frac{7}{192}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{179}{192}\right)\) | \(e\left(\frac{5}{24}\right)\) |
| \(\chi_{4608}(1111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{173}{192}\right)\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{145}{192}\right)\) | \(e\left(\frac{131}{192}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{31}{192}\right)\) | \(e\left(\frac{1}{24}\right)\) |
| \(\chi_{4608}(1159,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{192}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{107}{192}\right)\) | \(e\left(\frac{49}{192}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{31}{96}\right)\) | \(e\left(\frac{101}{192}\right)\) | \(e\left(\frac{11}{24}\right)\) |
| \(\chi_{4608}(1255,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{192}\right)\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{127}{192}\right)\) | \(e\left(\frac{173}{192}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{145}{192}\right)\) | \(e\left(\frac{7}{24}\right)\) |
| \(\chi_{4608}(1303,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{192}\right)\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{185}{192}\right)\) | \(e\left(\frac{187}{192}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{119}{192}\right)\) | \(e\left(\frac{17}{24}\right)\) |
| \(\chi_{4608}(1399,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{185}{192}\right)\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{13}{192}\right)\) | \(e\left(\frac{119}{192}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{31}{96}\right)\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{163}{192}\right)\) | \(e\left(\frac{13}{24}\right)\) |
| \(\chi_{4608}(1447,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{192}\right)\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{167}{192}\right)\) | \(e\left(\frac{37}{192}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{41}{192}\right)\) | \(e\left(\frac{23}{24}\right)\) |
| \(\chi_{4608}(1543,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{192}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{187}{192}\right)\) | \(e\left(\frac{161}{192}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{85}{192}\right)\) | \(e\left(\frac{19}{24}\right)\) |
| \(\chi_{4608}(1591,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{192}\right)\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{53}{192}\right)\) | \(e\left(\frac{175}{192}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{59}{192}\right)\) | \(e\left(\frac{5}{24}\right)\) |
| \(\chi_{4608}(1687,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{192}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{73}{192}\right)\) | \(e\left(\frac{107}{192}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{103}{192}\right)\) | \(e\left(\frac{1}{24}\right)\) |
| \(\chi_{4608}(1735,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{192}\right)\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{35}{192}\right)\) | \(e\left(\frac{25}{192}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{173}{192}\right)\) | \(e\left(\frac{11}{24}\right)\) |
| \(\chi_{4608}(1831,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{192}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{55}{192}\right)\) | \(e\left(\frac{149}{192}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{25}{192}\right)\) | \(e\left(\frac{7}{24}\right)\) |
| \(\chi_{4608}(1879,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{192}\right)\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{113}{192}\right)\) | \(e\left(\frac{163}{192}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{191}{192}\right)\) | \(e\left(\frac{17}{24}\right)\) |
| \(\chi_{4608}(1975,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{192}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{133}{192}\right)\) | \(e\left(\frac{95}{192}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{43}{192}\right)\) | \(e\left(\frac{13}{24}\right)\) |
| \(\chi_{4608}(2023,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{192}\right)\) | \(e\left(\frac{31}{96}\right)\) | \(e\left(\frac{95}{192}\right)\) | \(e\left(\frac{13}{192}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{113}{192}\right)\) | \(e\left(\frac{23}{24}\right)\) |
| \(\chi_{4608}(2119,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{192}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{115}{192}\right)\) | \(e\left(\frac{137}{192}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{157}{192}\right)\) | \(e\left(\frac{19}{24}\right)\) |
| \(\chi_{4608}(2167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{192}\right)\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{173}{192}\right)\) | \(e\left(\frac{151}{192}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{131}{192}\right)\) | \(e\left(\frac{5}{24}\right)\) |