from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([165,220,237]))
chi.galois_orbit()
[g,chi] = znchar(Mod(39,8470))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8470\) | |
| Conductor: | \(4235\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(330\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 4235.di | 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 | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8470}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{311}{330}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{71}{165}\right)\) | \(e\left(\frac{329}{330}\right)\) |
| \(\chi_{8470}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{16}{165}\right)\) | \(e\left(\frac{199}{330}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{271}{330}\right)\) |
| \(\chi_{8470}(149,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{139}{165}\right)\) | \(e\left(\frac{151}{330}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{61}{165}\right)\) | \(e\left(\frac{199}{330}\right)\) |
| \(\chi_{8470}(249,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{47}{165}\right)\) | \(e\left(\frac{203}{330}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{23}{165}\right)\) | \(e\left(\frac{167}{330}\right)\) |
| \(\chi_{8470}(359,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{223}{330}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{307}{330}\right)\) |
| \(\chi_{8470}(459,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{167}{330}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{62}{165}\right)\) | \(e\left(\frac{113}{330}\right)\) |
| \(\chi_{8470}(569,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{157}{330}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{43}{330}\right)\) |
| \(\chi_{8470}(739,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{179}{330}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{104}{165}\right)\) | \(e\left(\frac{131}{330}\right)\) |
| \(\chi_{8470}(809,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{71}{330}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{299}{330}\right)\) |
| \(\chi_{8470}(849,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{139}{330}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{181}{330}\right)\) |
| \(\chi_{8470}(919,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{241}{330}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{169}{330}\right)\) |
| \(\chi_{8470}(1019,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{32}{165}\right)\) | \(e\left(\frac{233}{330}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{47}{330}\right)\) |
| \(\chi_{8470}(1229,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{23}{165}\right)\) | \(e\left(\frac{317}{330}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{92}{165}\right)\) | \(e\left(\frac{173}{330}\right)\) |
| \(\chi_{8470}(1339,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{307}{330}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{103}{330}\right)\) |
| \(\chi_{8470}(1509,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{119}{330}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{41}{330}\right)\) |
| \(\chi_{8470}(1579,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{134}{165}\right)\) | \(e\left(\frac{161}{330}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{269}{330}\right)\) |
| \(\chi_{8470}(1619,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{76}{165}\right)\) | \(e\left(\frac{79}{330}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{139}{165}\right)\) | \(e\left(\frac{91}{330}\right)\) |
| \(\chi_{8470}(1689,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{49}{165}\right)\) | \(e\left(\frac{1}{330}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{31}{165}\right)\) | \(e\left(\frac{139}{330}\right)\) |
| \(\chi_{8470}(1789,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{17}{165}\right)\) | \(e\left(\frac{263}{330}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{68}{165}\right)\) | \(e\left(\frac{257}{330}\right)\) |
| \(\chi_{8470}(1899,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{283}{330}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{67}{330}\right)\) |
| \(\chi_{8470}(1999,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{137}{330}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{233}{330}\right)\) |
| \(\chi_{8470}(2109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{118}{165}\right)\) | \(e\left(\frac{127}{330}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{142}{165}\right)\) | \(e\left(\frac{163}{330}\right)\) |
| \(\chi_{8470}(2279,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{86}{165}\right)\) | \(e\left(\frac{59}{330}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{281}{330}\right)\) |
| \(\chi_{8470}(2349,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{251}{330}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{239}{330}\right)\) |
| \(\chi_{8470}(2389,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{106}{165}\right)\) | \(e\left(\frac{19}{330}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{1}{330}\right)\) |
| \(\chi_{8470}(2459,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{4}{165}\right)\) | \(e\left(\frac{91}{330}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{16}{165}\right)\) | \(e\left(\frac{109}{330}\right)\) |
| \(\chi_{8470}(2559,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{2}{165}\right)\) | \(e\left(\frac{293}{330}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{8}{165}\right)\) | \(e\left(\frac{137}{330}\right)\) |
| \(\chi_{8470}(2669,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{157}{165}\right)\) | \(e\left(\frac{313}{330}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{133}{165}\right)\) | \(e\left(\frac{277}{330}\right)\) |
| \(\chi_{8470}(2769,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{287}{330}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{293}{330}\right)\) |
| \(\chi_{8470}(2879,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{43}{165}\right)\) | \(e\left(\frac{277}{330}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{223}{330}\right)\) |
| \(\chi_{8470}(3049,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{329}{330}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{134}{165}\right)\) | \(e\left(\frac{191}{330}\right)\) |