from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2304, base_ring=CyclotomicField(96))
M = H._module
chi = DirichletCharacter(H, M([48,15,64]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,2304))
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.bu | 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: | $\Q(\zeta_{96})$ |
| Fixed field: | Number field defined by a degree 96 polynomial |
First 31 of 32 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2304}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{2304}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{2304}(151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{2304}(247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{2304}(295,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{2304}(391,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{96}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{2304}(439,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{2304}(535,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{2304}(583,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{2304}(679,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{2304}(727,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{2304}(823,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{2304}(871,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{31}{96}\right)\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{2304}(967,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{2304}(1015,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{2304}(1111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{2304}(1159,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{2304}(1255,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{2304}(1303,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{2304}(1399,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{2304}(1447,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{2304}(1543,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{2304}(1591,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{2304}(1687,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{2304}(1735,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{2304}(1831,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{2304}(1879,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{31}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{2304}(1975,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{31}{96}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{2304}(2023,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{2304}(2119,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{2304}(2167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |