from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8015, base_ring=CyclotomicField(228))
M = H._module
chi = DirichletCharacter(H, M([171,114,64]))
chi.galois_orbit()
[g,chi] = znchar(Mod(48,8015))
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: | \(228\) | 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: | $\Q(\zeta_{228})$ |
| Fixed field: | Number field defined by a degree 228 polynomial (not computed) |
First 31 of 72 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8015}(48,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{76}\right)\) | \(e\left(\frac{31}{228}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{89}{114}\right)\) | \(e\left(\frac{71}{76}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{97}{228}\right)\) | \(e\left(\frac{45}{76}\right)\) | \(e\left(\frac{11}{19}\right)\) |
| \(\chi_{8015}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{76}\right)\) | \(e\left(\frac{163}{228}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{35}{76}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{157}{228}\right)\) | \(e\left(\frac{65}{76}\right)\) | \(e\left(\frac{18}{19}\right)\) |
| \(\chi_{8015}(132,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{76}\right)\) | \(e\left(\frac{25}{228}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{45}{76}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{115}{228}\right)\) | \(e\left(\frac{51}{76}\right)\) | \(e\left(\frac{15}{19}\right)\) |
| \(\chi_{8015}(153,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{76}\right)\) | \(e\left(\frac{227}{228}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{59}{76}\right)\) | \(e\left(\frac{113}{114}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{41}{228}\right)\) | \(e\left(\frac{1}{76}\right)\) | \(e\left(\frac{7}{19}\right)\) |
| \(\chi_{8015}(167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{76}\right)\) | \(e\left(\frac{197}{228}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{131}{228}\right)\) | \(e\left(\frac{31}{76}\right)\) | \(e\left(\frac{8}{19}\right)\) |
| \(\chi_{8015}(412,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{76}\right)\) | \(e\left(\frac{193}{228}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{13}{76}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{67}{228}\right)\) | \(e\left(\frac{35}{76}\right)\) | \(e\left(\frac{17}{19}\right)\) |
| \(\chi_{8015}(587,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{76}\right)\) | \(e\left(\frac{85}{228}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{1}{76}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{163}{228}\right)\) | \(e\left(\frac{67}{76}\right)\) | \(e\left(\frac{13}{19}\right)\) |
| \(\chi_{8015}(762,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{76}\right)\) | \(e\left(\frac{185}{228}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{29}{76}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{167}{228}\right)\) | \(e\left(\frac{43}{76}\right)\) | \(e\left(\frac{16}{19}\right)\) |
| \(\chi_{8015}(867,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{76}\right)\) | \(e\left(\frac{109}{228}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{29}{76}\right)\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{91}{228}\right)\) | \(e\left(\frac{43}{76}\right)\) | \(e\left(\frac{16}{19}\right)\) |
| \(\chi_{8015}(1007,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{76}\right)\) | \(e\left(\frac{137}{228}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{55}{114}\right)\) | \(e\left(\frac{49}{76}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{83}{228}\right)\) | \(e\left(\frac{15}{76}\right)\) | \(e\left(\frac{10}{19}\right)\) |
| \(\chi_{8015}(1042,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{76}\right)\) | \(e\left(\frac{121}{228}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{55}{228}\right)\) | \(e\left(\frac{31}{76}\right)\) | \(e\left(\frac{8}{19}\right)\) |
| \(\chi_{8015}(1112,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{76}\right)\) | \(e\left(\frac{181}{228}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{37}{76}\right)\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{103}{228}\right)\) | \(e\left(\frac{47}{76}\right)\) | \(e\left(\frac{6}{19}\right)\) |
| \(\chi_{8015}(1133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{76}\right)\) | \(e\left(\frac{187}{228}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{63}{76}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{85}{228}\right)\) | \(e\left(\frac{41}{76}\right)\) | \(e\left(\frac{2}{19}\right)\) |
| \(\chi_{8015}(1182,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{76}\right)\) | \(e\left(\frac{217}{228}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{41}{76}\right)\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{223}{228}\right)\) | \(e\left(\frac{11}{76}\right)\) | \(e\left(\frac{1}{19}\right)\) |
| \(\chi_{8015}(1518,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{76}\right)\) | \(e\left(\frac{203}{228}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{31}{76}\right)\) | \(e\left(\frac{89}{114}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{113}{228}\right)\) | \(e\left(\frac{25}{76}\right)\) | \(e\left(\frac{4}{19}\right)\) |
| \(\chi_{8015}(1532,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{76}\right)\) | \(e\left(\frac{29}{228}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{37}{76}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{179}{228}\right)\) | \(e\left(\frac{47}{76}\right)\) | \(e\left(\frac{6}{19}\right)\) |
| \(\chi_{8015}(1567,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{76}\right)\) | \(e\left(\frac{17}{228}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{61}{76}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{215}{228}\right)\) | \(e\left(\frac{59}{76}\right)\) | \(e\left(\frac{14}{19}\right)\) |
| \(\chi_{8015}(1623,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{76}\right)\) | \(e\left(\frac{107}{228}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{71}{76}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{173}{228}\right)\) | \(e\left(\frac{45}{76}\right)\) | \(e\left(\frac{11}{19}\right)\) |
| \(\chi_{8015}(1658,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{76}\right)\) | \(e\left(\frac{215}{228}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{7}{76}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{77}{228}\right)\) | \(e\left(\frac{13}{76}\right)\) | \(e\left(\frac{15}{19}\right)\) |
| \(\chi_{8015}(2232,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{76}\right)\) | \(e\left(\frac{157}{228}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{9}{76}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{175}{228}\right)\) | \(e\left(\frac{71}{76}\right)\) | \(e\left(\frac{3}{19}\right)\) |
| \(\chi_{8015}(2372,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{76}\right)\) | \(e\left(\frac{53}{228}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{65}{76}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{107}{228}\right)\) | \(e\left(\frac{23}{76}\right)\) | \(e\left(\frac{9}{19}\right)\) |
| \(\chi_{8015}(2463,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{76}\right)\) | \(e\left(\frac{191}{228}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{55}{76}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{149}{228}\right)\) | \(e\left(\frac{37}{76}\right)\) | \(e\left(\frac{12}{19}\right)\) |
| \(\chi_{8015}(2533,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{76}\right)\) | \(e\left(\frac{119}{228}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{47}{76}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{137}{228}\right)\) | \(e\left(\frac{33}{76}\right)\) | \(e\left(\frac{3}{19}\right)\) |
| \(\chi_{8015}(2743,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{76}\right)\) | \(e\left(\frac{35}{228}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{63}{76}\right)\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{161}{228}\right)\) | \(e\left(\frac{41}{76}\right)\) | \(e\left(\frac{2}{19}\right)\) |
| \(\chi_{8015}(2757,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{76}\right)\) | \(e\left(\frac{173}{228}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{53}{76}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{203}{228}\right)\) | \(e\left(\frac{55}{76}\right)\) | \(e\left(\frac{5}{19}\right)\) |
| \(\chi_{8015}(2897,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{76}\right)\) | \(e\left(\frac{65}{228}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{41}{76}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{71}{228}\right)\) | \(e\left(\frac{11}{76}\right)\) | \(e\left(\frac{1}{19}\right)\) |
| \(\chi_{8015}(2932,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{76}\right)\) | \(e\left(\frac{37}{228}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{21}{76}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{79}{228}\right)\) | \(e\left(\frac{39}{76}\right)\) | \(e\left(\frac{7}{19}\right)\) |
| \(\chi_{8015}(3002,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{76}\right)\) | \(e\left(\frac{13}{228}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{69}{76}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{151}{228}\right)\) | \(e\left(\frac{63}{76}\right)\) | \(e\left(\frac{4}{19}\right)\) |
| \(\chi_{8015}(3058,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{76}\right)\) | \(e\left(\frac{175}{228}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{11}{76}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{121}{228}\right)\) | \(e\left(\frac{53}{76}\right)\) | \(e\left(\frac{10}{19}\right)\) |
| \(\chi_{8015}(3107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{75}{76}\right)\) | \(e\left(\frac{125}{228}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{73}{76}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{119}{228}\right)\) | \(e\left(\frac{27}{76}\right)\) | \(e\left(\frac{18}{19}\right)\) |
| \(\chi_{8015}(3128,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{76}\right)\) | \(e\left(\frac{91}{228}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{27}{76}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{145}{228}\right)\) | \(e\left(\frac{61}{76}\right)\) | \(e\left(\frac{9}{19}\right)\) |