from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(291, base_ring=CyclotomicField(96))
M = H._module
chi = DirichletCharacter(H, M([48,1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,291))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(291\) | |
| Conductor: | \(291\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(96\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| 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 | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{291}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{31}{96}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{291}(14,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{291}(17,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{291}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{291}(26,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{291}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{291}(38,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{291}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{291}(56,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{291}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{291}(68,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{291}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{291}(74,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{291}(80,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{291}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{291}(92,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{291}(104,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{291}(107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{291}(110,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{291}(134,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{291}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{291}(155,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{31}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{291}(173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{31}{96}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{291}(179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{291}(209,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{291}(215,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{291}(233,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{79}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{291}(251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{31}{96}\right)\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{291}(254,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{291}(278,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{291}(281,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{7}{12}\right)\) |