from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4235, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([165,220,6]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,4235))
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: | 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 330 polynomial (not computed) |
First 31 of 80 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4235}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{281}{330}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{67}{165}\right)\) | \(e\left(\frac{19}{330}\right)\) |
| \(\chi_{4235}(114,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{241}{330}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{76}{165}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{149}{330}\right)\) |
| \(\chi_{4235}(179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{329}{330}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{163}{165}\right)\) | \(e\left(\frac{61}{330}\right)\) |
| \(\chi_{4235}(214,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{330}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{223}{330}\right)\) |
| \(\chi_{4235}(284,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{233}{330}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{68}{165}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{307}{330}\right)\) |
| \(\chi_{4235}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{330}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{161}{330}\right)\) |
| \(\chi_{4235}(324,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{330}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{127}{165}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{173}{330}\right)\) |
| \(\chi_{4235}(389,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{330}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{86}{165}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{199}{330}\right)\) |
| \(\chi_{4235}(394,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{313}{330}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{47}{330}\right)\) |
| \(\chi_{4235}(499,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{211}{330}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{92}{165}\right)\) | \(e\left(\frac{329}{330}\right)\) |
| \(\chi_{4235}(564,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{330}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{301}{330}\right)\) |
| \(\chi_{4235}(599,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{330}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{47}{165}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{103}{330}\right)\) |
| \(\chi_{4235}(669,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{293}{330}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{277}{330}\right)\) |
| \(\chi_{4235}(674,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{330}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{34}{165}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{68}{165}\right)\) | \(e\left(\frac{71}{330}\right)\) |
| \(\chi_{4235}(709,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{330}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{74}{165}\right)\) | \(e\left(\frac{53}{330}\right)\) |
| \(\chi_{4235}(774,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{221}{330}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{49}{330}\right)\) |
| \(\chi_{4235}(779,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{330}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{43}{165}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{86}{165}\right)\) | \(e\left(\frac{17}{330}\right)\) |
| \(\chi_{4235}(884,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{181}{330}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{16}{165}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{32}{165}\right)\) | \(e\left(\frac{179}{330}\right)\) |
| \(\chi_{4235}(949,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{330}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{29}{165}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{58}{165}\right)\) | \(e\left(\frac{211}{330}\right)\) |
| \(\chi_{4235}(984,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{287}{330}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{313}{330}\right)\) |
| \(\chi_{4235}(1054,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{330}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{23}{165}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{247}{330}\right)\) |
| \(\chi_{4235}(1059,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{330}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{49}{165}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{311}{330}\right)\) |
| \(\chi_{4235}(1094,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{277}{330}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{263}{330}\right)\) |
| \(\chi_{4235}(1159,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{330}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{229}{330}\right)\) |
| \(\chi_{4235}(1164,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{330}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{317}{330}\right)\) |
| \(\chi_{4235}(1269,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{330}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{151}{165}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{29}{330}\right)\) |
| \(\chi_{4235}(1369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{197}{330}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{32}{165}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{193}{330}\right)\) |
| \(\chi_{4235}(1439,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{330}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{83}{165}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{1}{165}\right)\) | \(e\left(\frac{217}{330}\right)\) |
| \(\chi_{4235}(1444,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{330}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{221}{330}\right)\) |
| \(\chi_{4235}(1544,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{330}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{161}{165}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{157}{165}\right)\) | \(e\left(\frac{79}{330}\right)\) |
| \(\chi_{4235}(1549,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{330}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{163}{165}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{161}{165}\right)\) | \(e\left(\frac{287}{330}\right)\) |