from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7728, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([66,33,66,88,54]))
chi.galois_orbit()
[g,chi] = znchar(Mod(11,7728))
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 | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{7728}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{132}\right)\) | \(e\left(\frac{79}{132}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{23}{132}\right)\) | \(e\left(\frac{10}{11}\right)\) |
| \(\chi_{7728}(107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{132}\right)\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{67}{132}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{3}{11}\right)\) |
| \(\chi_{7728}(779,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{91}{132}\right)\) | \(e\left(\frac{8}{11}\right)\) |
| \(\chi_{7728}(1019,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{91}{132}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{4}{11}\right)\) |
| \(\chi_{7728}(1115,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{132}\right)\) | \(e\left(\frac{35}{132}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{67}{132}\right)\) | \(e\left(\frac{10}{11}\right)\) |
| \(\chi_{7728}(1187,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{5}{11}\right)\) |
| \(\chi_{7728}(1355,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{132}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{131}{132}\right)\) | \(e\left(\frac{1}{11}\right)\) |
| \(\chi_{7728}(1523,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{109}{132}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{5}{132}\right)\) | \(e\left(\frac{6}{11}\right)\) |
| \(\chi_{7728}(1859,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{132}\right)\) | \(e\left(\frac{73}{132}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{2}{11}\right)\) |
| \(\chi_{7728}(2123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{132}\right)\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{115}{132}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{4}{11}\right)\) |
| \(\chi_{7728}(2195,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{7}{11}\right)\) |
| \(\chi_{7728}(2291,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{132}\right)\) | \(e\left(\frac{89}{132}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{61}{132}\right)\) | \(e\left(\frac{5}{11}\right)\) |
| \(\chi_{7728}(2363,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{103}{132}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{35}{132}\right)\) | \(e\left(\frac{9}{11}\right)\) |
| \(\chi_{7728}(2459,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{119}{132}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{103}{132}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{1}{11}\right)\) |
| \(\chi_{7728}(2627,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{132}\right)\) | \(e\left(\frac{65}{132}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{6}{11}\right)\) |
| \(\chi_{7728}(2867,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{89}{132}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{3}{11}\right)\) |
| \(\chi_{7728}(2963,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{132}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{2}{11}\right)\) |
| \(\chi_{7728}(3299,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{61}{132}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{37}{132}\right)\) | \(e\left(\frac{7}{11}\right)\) |
| \(\chi_{7728}(3467,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{91}{132}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{79}{132}\right)\) | \(e\left(\frac{9}{11}\right)\) |
| \(\chi_{7728}(3539,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{132}\right)\) | \(e\left(\frac{61}{132}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{65}{132}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{113}{132}\right)\) | \(e\left(\frac{8}{11}\right)\) |
| \(\chi_{7728}(3875,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{132}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{89}{132}\right)\) | \(e\left(\frac{10}{11}\right)\) |
| \(\chi_{7728}(3971,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{132}\right)\) | \(e\left(\frac{5}{132}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{3}{11}\right)\) |
| \(\chi_{7728}(4643,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{109}{132}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{25}{132}\right)\) | \(e\left(\frac{8}{11}\right)\) |
| \(\chi_{7728}(4883,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{25}{132}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{5}{132}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{4}{11}\right)\) |
| \(\chi_{7728}(4979,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{73}{132}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{10}{11}\right)\) |
| \(\chi_{7728}(5051,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{132}\right)\) | \(e\left(\frac{67}{132}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{119}{132}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{5}{11}\right)\) |
| \(\chi_{7728}(5219,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{132}\right)\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{65}{132}\right)\) | \(e\left(\frac{1}{11}\right)\) |
| \(\chi_{7728}(5387,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{35}{132}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{6}{11}\right)\) |
| \(\chi_{7728}(5723,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{107}{132}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{119}{132}\right)\) | \(e\left(\frac{2}{11}\right)\) |
| \(\chi_{7728}(5987,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{132}\right)\) | \(e\left(\frac{113}{132}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{73}{132}\right)\) | \(e\left(\frac{4}{11}\right)\) |
| \(\chi_{7728}(6059,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{7}{11}\right)\) |