from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7728, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([0,33,66,110,6]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,7728))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(7728\) | |
| Conductor: | \(7728\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(132\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{132})$ |
| Fixed field: | Number field defined by a degree 132 polynomial (not computed) |
First 31 of 40 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{7728}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{132}\right)\) | \(e\left(\frac{65}{132}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{79}{132}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{115}{132}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{7728}(341,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{132}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{7728}(605,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{91}{132}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{5}{132}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{7728}(677,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{127}{132}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{103}{132}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{7728}(773,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{119}{132}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{7728}(845,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{25}{132}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{7728}(941,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{132}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{65}{132}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{7728}(1109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{109}{132}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{35}{132}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{7728}(1349,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{132}\right)\) | \(e\left(\frac{5}{132}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{67}{132}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{7728}(1445,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{132}\right)\) | \(e\left(\frac{73}{132}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{107}{132}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{119}{132}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{7728}(1781,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{7728}(1949,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{103}{132}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{113}{132}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{7728}(2021,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{91}{132}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{7728}(2357,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{67}{132}\right)\) | \(e\left(\frac{9}{22}\right)\) |
| \(\chi_{7728}(2453,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{23}{132}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{107}{132}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{7728}(3125,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{132}\right)\) | \(e\left(\frac{61}{132}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{131}{132}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{7728}(3365,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{132}\right)\) | \(e\left(\frac{113}{132}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{115}{132}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{7728}(3461,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{132}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{23}{132}\right)\) | \(e\left(\frac{9}{22}\right)\) |
| \(\chi_{7728}(3533,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{23}{132}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{61}{132}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{7728}(3701,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{132}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{103}{132}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{7728}(3869,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{132}\right)\) | \(e\left(\frac{131}{132}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{7728}(4205,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{7728}(4469,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{25}{132}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{7728}(4541,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{132}\right)\) | \(e\left(\frac{107}{132}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{61}{132}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{37}{132}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{7728}(4637,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{132}\right)\) | \(e\left(\frac{67}{132}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{7728}(4709,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{91}{132}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{79}{132}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{7728}(4805,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{132}\right)\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{131}{132}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{7728}(4973,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{5}{132}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{7728}(5213,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{132}\right)\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{7728}(5309,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{7728}(5645,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{3}{22}\right)\) |