from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2304, base_ring=CyclotomicField(96))
M = H._module
chi = DirichletCharacter(H, M([48,21,80]))
chi.galois_orbit()
[g,chi] = znchar(Mod(23,2304))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2304\) | |
| Conductor: | \(1152\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(96\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1152.bs | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{96})$ |
| Fixed field: | Number field defined by a degree 96 polynomial |
First 31 of 32 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2304}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{2304}(119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{2304}(167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{2304}(263,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{2304}(311,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{2304}(407,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{2304}(455,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{2304}(551,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{2304}(599,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{2304}(695,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{2304}(743,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{2304}(839,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{2304}(887,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{2304}(983,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{31}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{2304}(1031,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{96}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{2304}(1127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{2304}(1175,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{2304}(1271,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{2304}(1319,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{2304}(1415,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{2304}(1463,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{31}{96}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{2304}(1559,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{2304}(1607,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{2304}(1703,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{2304}(1751,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{2304}(1847,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{2304}(1895,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{2304}(1991,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{2304}(2039,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{2304}(2135,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{2304}(2183,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |