from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4235, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([0,110,12]))
chi.galois_orbit()
[g,chi] = znchar(Mod(16,4235))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4235\) | |
| Conductor: | \(847\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(165\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 847.bc | 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: | $\Q(\zeta_{165})$ |
| Fixed field: | Number field defined by a degree 165 polynomial (not computed) |
First 31 of 80 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4235}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{67}{165}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{134}{165}\right)\) | \(e\left(\frac{19}{165}\right)\) |
| \(\chi_{4235}(86,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{165}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{133}{165}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{101}{165}\right)\) | \(e\left(\frac{151}{165}\right)\) |
| \(\chi_{4235}(191,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{68}{165}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{107}{165}\right)\) | \(e\left(\frac{142}{165}\right)\) |
| \(\chi_{4235}(256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{165}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{134}{165}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{38}{165}\right)\) |
| \(\chi_{4235}(291,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{76}{165}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{139}{165}\right)\) | \(e\left(\frac{149}{165}\right)\) |
| \(\chi_{4235}(361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{165}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{61}{165}\right)\) | \(e\left(\frac{101}{165}\right)\) |
| \(\chi_{4235}(401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{165}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{142}{165}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{119}{165}\right)\) | \(e\left(\frac{124}{165}\right)\) |
| \(\chi_{4235}(466,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{165}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{146}{165}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{127}{165}\right)\) | \(e\left(\frac{2}{165}\right)\) |
| \(\chi_{4235}(471,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{136}{165}\right)\) |
| \(\chi_{4235}(576,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{165}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{106}{165}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{47}{165}\right)\) | \(e\left(\frac{67}{165}\right)\) |
| \(\chi_{4235}(641,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{157}{165}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{149}{165}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{133}{165}\right)\) | \(e\left(\frac{158}{165}\right)\) |
| \(\chi_{4235}(676,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{165}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{62}{165}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{124}{165}\right)\) | \(e\left(\frac{89}{165}\right)\) |
| \(\chi_{4235}(746,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{165}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{8}{165}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{16}{165}\right)\) | \(e\left(\frac{86}{165}\right)\) |
| \(\chi_{4235}(751,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{165}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{4}{165}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{8}{165}\right)\) | \(e\left(\frac{43}{165}\right)\) |
| \(\chi_{4235}(786,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{104}{165}\right)\) | \(e\left(\frac{64}{165}\right)\) |
| \(\chi_{4235}(851,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{165}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{67}{165}\right)\) | \(e\left(\frac{92}{165}\right)\) |
| \(\chi_{4235}(961,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{76}{165}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{157}{165}\right)\) |
| \(\chi_{4235}(1026,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{163}{165}\right)\) | \(e\left(\frac{113}{165}\right)\) |
| \(\chi_{4235}(1061,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{165}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{29}{165}\right)\) |
| \(\chi_{4235}(1131,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{165}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{68}{165}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{71}{165}\right)\) |
| \(\chi_{4235}(1136,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{92}{165}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{163}{165}\right)\) |
| \(\chi_{4235}(1171,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{146}{165}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{127}{165}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{4}{165}\right)\) |
| \(\chi_{4235}(1236,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{165}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{86}{165}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{17}{165}\right)\) |
| \(\chi_{4235}(1241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{74}{165}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{106}{165}\right)\) |
| \(\chi_{4235}(1346,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{165}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{92}{165}\right)\) | \(e\left(\frac{82}{165}\right)\) |
| \(\chi_{4235}(1411,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{68}{165}\right)\) |
| \(\chi_{4235}(1446,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{106}{165}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{47}{165}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{134}{165}\right)\) |
| \(\chi_{4235}(1516,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{56}{165}\right)\) |
| \(\chi_{4235}(1521,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{165}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{34}{165}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{68}{165}\right)\) | \(e\left(\frac{118}{165}\right)\) |
| \(\chi_{4235}(1556,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{165}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{74}{165}\right)\) | \(e\left(\frac{109}{165}\right)\) |
| \(\chi_{4235}(1621,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{107}{165}\right)\) |