from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([0,275,327]))
chi.galois_orbit()
[g,chi] = znchar(Mod(61,8470))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8470\) | |
| Conductor: | \(847\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(330\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 847.be | 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 | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8470}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{68}{165}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{17}{330}\right)\) | \(e\left(\frac{47}{165}\right)\) |
| \(\chi_{8470}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{86}{165}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{193}{330}\right)\) | \(e\left(\frac{58}{165}\right)\) |
| \(\chi_{8470}(171,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{43}{165}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{217}{330}\right)\) | \(e\left(\frac{37}{165}\right)\) |
| \(\chi_{8470}(271,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{67}{165}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{41}{330}\right)\) | \(e\left(\frac{26}{165}\right)\) |
| \(\chi_{8470}(381,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{2}{165}\right)\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{181}{330}\right)\) | \(e\left(\frac{151}{165}\right)\) |
| \(\chi_{8470}(591,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{61}{165}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{139}{330}\right)\) | \(e\left(\frac{64}{165}\right)\) |
| \(\chi_{8470}(761,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{151}{165}\right)\) | \(e\left(\frac{47}{165}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{53}{330}\right)\) | \(e\left(\frac{98}{165}\right)\) |
| \(\chi_{8470}(831,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{317}{330}\right)\) | \(e\left(\frac{32}{165}\right)\) |
| \(\chi_{8470}(871,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{103}{330}\right)\) | \(e\left(\frac{13}{165}\right)\) |
| \(\chi_{8470}(1041,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{251}{330}\right)\) | \(e\left(\frac{131}{165}\right)\) |
| \(\chi_{8470}(1151,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{61}{330}\right)\) | \(e\left(\frac{91}{165}\right)\) |
| \(\chi_{8470}(1251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{13}{165}\right)\) | \(e\left(\frac{86}{165}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{269}{330}\right)\) | \(e\left(\frac{74}{165}\right)\) |
| \(\chi_{8470}(1361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{199}{330}\right)\) | \(e\left(\frac{94}{165}\right)\) |
| \(\chi_{8470}(1531,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{16}{165}\right)\) | \(e\left(\frac{17}{165}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{293}{330}\right)\) | \(e\left(\frac{53}{165}\right)\) |
| \(\chi_{8470}(1601,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{139}{165}\right)\) | \(e\left(\frac{158}{165}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{287}{330}\right)\) | \(e\left(\frac{17}{165}\right)\) |
| \(\chi_{8470}(1641,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{146}{165}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{13}{330}\right)\) | \(e\left(\frac{133}{165}\right)\) |
| \(\chi_{8470}(1711,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{133}{165}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{157}{330}\right)\) | \(e\left(\frac{7}{165}\right)\) |
| \(\chi_{8470}(1811,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{29}{165}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{131}{330}\right)\) | \(e\left(\frac{71}{165}\right)\) |
| \(\chi_{8470}(1921,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{271}{330}\right)\) | \(e\left(\frac{31}{165}\right)\) |
| \(\chi_{8470}(2021,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{161}{165}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{329}{330}\right)\) | \(e\left(\frac{104}{165}\right)\) |
| \(\chi_{8470}(2131,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{53}{165}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{259}{330}\right)\) | \(e\left(\frac{124}{165}\right)\) |
| \(\chi_{8470}(2301,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{203}{330}\right)\) | \(e\left(\frac{8}{165}\right)\) |
| \(\chi_{8470}(2371,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{257}{330}\right)\) | \(e\left(\frac{2}{165}\right)\) |
| \(\chi_{8470}(2481,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{119}{165}\right)\) | \(e\left(\frac{13}{165}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{127}{330}\right)\) | \(e\left(\frac{157}{165}\right)\) |
| \(\chi_{8470}(2691,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{151}{330}\right)\) | \(e\left(\frac{136}{165}\right)\) |
| \(\chi_{8470}(2791,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{71}{165}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{59}{330}\right)\) | \(e\left(\frac{134}{165}\right)\) |
| \(\chi_{8470}(3071,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{76}{165}\right)\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{113}{330}\right)\) | \(e\left(\frac{128}{165}\right)\) |
| \(\chi_{8470}(3141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{49}{165}\right)\) | \(e\left(\frac{83}{165}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{227}{330}\right)\) | \(e\left(\frac{152}{165}\right)\) |
| \(\chi_{8470}(3181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{157}{165}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{163}{330}\right)\) | \(e\left(\frac{43}{165}\right)\) |
| \(\chi_{8470}(3251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{74}{165}\right)\) | \(e\left(\frac{58}{165}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{97}{330}\right)\) | \(e\left(\frac{142}{165}\right)\) |
| \(\chi_{8470}(3351,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{221}{330}\right)\) | \(e\left(\frac{116}{165}\right)\) |