from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8015, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([57,38,102]))
chi.galois_orbit()
[g,chi] = znchar(Mod(44,8015))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8015\) | |
| Conductor: | \(8015\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(114\) | 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: | $\Q(\zeta_{57})$ |
| Fixed field: | Number field defined by a degree 114 polynomial (not computed) |
First 31 of 36 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8015}(44,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{47}{57}\right)\) |
| \(\chi_{8015}(214,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{4}{57}\right)\) |
| \(\chi_{8015}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{50}{57}\right)\) |
| \(\chi_{8015}(394,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{89}{114}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{5}{57}\right)\) |
| \(\chi_{8015}(704,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{25}{57}\right)\) |
| \(\chi_{8015}(744,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{11}{57}\right)\) |
| \(\chi_{8015}(1619,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{35}{57}\right)\) |
| \(\chi_{8015}(1724,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{2}{57}\right)\) |
| \(\chi_{8015}(1859,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{13}{57}\right)\) |
| \(\chi_{8015}(2104,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{22}{57}\right)\) |
| \(\chi_{8015}(2279,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{1}{57}\right)\) |
| \(\chi_{8015}(2809,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{114}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{53}{57}\right)\) |
| \(\chi_{8015}(2909,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{52}{57}\right)\) |
| \(\chi_{8015}(3019,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{26}{57}\right)\) |
| \(\chi_{8015}(3259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{114}\right)\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{55}{57}\right)\) |
| \(\chi_{8015}(3539,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{37}{57}\right)\) |
| \(\chi_{8015}(3649,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{113}{114}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{23}{57}\right)\) |
| \(\chi_{8015}(3889,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{55}{114}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{46}{57}\right)\) |
| \(\chi_{8015}(4139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{44}{57}\right)\) |
| \(\chi_{8015}(4554,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{10}{57}\right)\) |
| \(\chi_{8015}(4624,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{28}{57}\right)\) |
| \(\chi_{8015}(4869,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{31}{57}\right)\) |
| \(\chi_{8015}(4974,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{113}{114}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{43}{57}\right)\) |
| \(\chi_{8015}(5294,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{32}{57}\right)\) |
| \(\chi_{8015}(5324,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{89}{114}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{49}{57}\right)\) |
| \(\chi_{8015}(5539,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{41}{57}\right)\) |
| \(\chi_{8015}(5714,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{20}{57}\right)\) |
| \(\chi_{8015}(6199,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{16}{57}\right)\) |
| \(\chi_{8015}(6304,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{40}{57}\right)\) |
| \(\chi_{8015}(6344,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{14}{57}\right)\) |
| \(\chi_{8015}(6694,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{17}{57}\right)\) |