from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4235, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([165,220,237]))
chi.galois_orbit()
[g,chi] = znchar(Mod(39,4235))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4235\) | |
| Conductor: | \(4235\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(330\) | 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: | $\Q(\zeta_{165})$ |
| Fixed field: | Number field defined by a degree 330 polynomial (not computed) |
First 31 of 80 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4235}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{17}{165}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{34}{165}\right)\) | \(e\left(\frac{59}{165}\right)\) |
| \(\chi_{4235}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{74}{165}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{53}{165}\right)\) |
| \(\chi_{4235}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{165}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{16}{165}\right)\) |
| \(\chi_{4235}(149,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{139}{165}\right)\) |
| \(\chi_{4235}(184,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{53}{165}\right)\) | \(e\left(\frac{58}{165}\right)\) |
| \(\chi_{4235}(249,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{97}{165}\right)\) | \(e\left(\frac{47}{165}\right)\) |
| \(\chi_{4235}(359,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{61}{165}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{37}{165}\right)\) |
| \(\chi_{4235}(424,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{107}{165}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{49}{165}\right)\) | \(e\left(\frac{119}{165}\right)\) |
| \(\chi_{4235}(459,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{118}{165}\right)\) | \(e\left(\frac{98}{165}\right)\) |
| \(\chi_{4235}(464,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{118}{165}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{71}{165}\right)\) | \(e\left(\frac{31}{165}\right)\) |
| \(\chi_{4235}(534,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{86}{165}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{34}{165}\right)\) |
| \(\chi_{4235}(569,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{165}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{23}{165}\right)\) | \(e\left(\frac{103}{165}\right)\) |
| \(\chi_{4235}(634,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{165}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{161}{165}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{157}{165}\right)\) | \(e\left(\frac{122}{165}\right)\) |
| \(\chi_{4235}(739,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{124}{165}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{83}{165}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{1}{165}\right)\) | \(e\left(\frac{26}{165}\right)\) |
| \(\chi_{4235}(744,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{17}{165}\right)\) | \(e\left(\frac{112}{165}\right)\) |
| \(\chi_{4235}(809,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{165}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{32}{165}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{14}{165}\right)\) |
| \(\chi_{4235}(849,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{165}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{58}{165}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{46}{165}\right)\) |
| \(\chi_{4235}(919,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{97}{165}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{29}{165}\right)\) | \(e\left(\frac{94}{165}\right)\) |
| \(\chi_{4235}(954,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{158}{165}\right)\) | \(e\left(\frac{148}{165}\right)\) |
| \(\chi_{4235}(1019,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{165}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{32}{165}\right)\) |
| \(\chi_{4235}(1124,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{23}{165}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{41}{165}\right)\) |
| \(\chi_{4235}(1194,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{165}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{74}{165}\right)\) |
| \(\chi_{4235}(1229,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{165}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{29}{165}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{58}{165}\right)\) | \(e\left(\frac{23}{165}\right)\) |
| \(\chi_{4235}(1234,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{163}{165}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{161}{165}\right)\) | \(e\left(\frac{61}{165}\right)\) |
| \(\chi_{4235}(1339,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{32}{165}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{28}{165}\right)\) |
| \(\chi_{4235}(1404,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{107}{165}\right)\) |
| \(\chi_{4235}(1509,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{56}{165}\right)\) |
| \(\chi_{4235}(1514,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{158}{165}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{151}{165}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{97}{165}\right)\) |
| \(\chi_{4235}(1579,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{106}{165}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{47}{165}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{134}{165}\right)\) |
| \(\chi_{4235}(1614,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{68}{165}\right)\) |
| \(\chi_{4235}(1619,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{134}{165}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{76}{165}\right)\) |