from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([165,275,249]))
chi.galois_orbit()
[g,chi] = znchar(Mod(19,8470))
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.dj | 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 | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8470}(19,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{101}{330}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{239}{330}\right)\) | \(e\left(\frac{283}{330}\right)\) |
| \(\chi_{8470}(129,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{1}{330}\right)\) | \(e\left(\frac{16}{165}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{169}{330}\right)\) | \(e\left(\frac{323}{330}\right)\) |
| \(\chi_{8470}(299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{167}{330}\right)\) | \(e\left(\frac{32}{165}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{173}{330}\right)\) | \(e\left(\frac{151}{330}\right)\) |
| \(\chi_{8470}(369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{323}{330}\right)\) | \(e\left(\frac{53}{165}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{137}{330}\right)\) | \(e\left(\frac{49}{330}\right)\) |
| \(\chi_{8470}(409,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{97}{330}\right)\) | \(e\left(\frac{67}{165}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{223}{330}\right)\) | \(e\left(\frac{311}{330}\right)\) |
| \(\chi_{8470}(479,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{43}{330}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{7}{330}\right)\) | \(e\left(\frac{29}{330}\right)\) |
| \(\chi_{8470}(579,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{89}{330}\right)\) | \(e\left(\frac{104}{165}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{191}{330}\right)\) | \(e\left(\frac{37}{330}\right)\) |
| \(\chi_{8470}(689,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{289}{330}\right)\) | \(e\left(\frac{4}{165}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{1}{330}\right)\) | \(e\left(\frac{287}{330}\right)\) |
| \(\chi_{8470}(789,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{281}{330}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{299}{330}\right)\) | \(e\left(\frac{13}{330}\right)\) |
| \(\chi_{8470}(899,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{181}{330}\right)\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{229}{330}\right)\) | \(e\left(\frac{53}{330}\right)\) |
| \(\chi_{8470}(1069,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{227}{330}\right)\) | \(e\left(\frac{2}{165}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{83}{330}\right)\) | \(e\left(\frac{61}{330}\right)\) |
| \(\chi_{8470}(1139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{233}{330}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{107}{330}\right)\) | \(e\left(\frac{19}{330}\right)\) |
| \(\chi_{8470}(1179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{157}{330}\right)\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{133}{330}\right)\) | \(e\left(\frac{221}{330}\right)\) |
| \(\chi_{8470}(1249,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{283}{330}\right)\) | \(e\left(\frac{73}{165}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{307}{330}\right)\) | \(e\left(\frac{329}{330}\right)\) |
| \(\chi_{8470}(1349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{59}{330}\right)\) | \(e\left(\frac{119}{165}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{71}{330}\right)\) | \(e\left(\frac{247}{330}\right)\) |
| \(\chi_{8470}(1459,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{259}{330}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{211}{330}\right)\) | \(e\left(\frac{167}{330}\right)\) |
| \(\chi_{8470}(1559,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{131}{330}\right)\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{29}{330}\right)\) | \(e\left(\frac{73}{330}\right)\) |
| \(\chi_{8470}(1669,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{31}{330}\right)\) | \(e\left(\frac{1}{165}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{289}{330}\right)\) | \(e\left(\frac{113}{330}\right)\) |
| \(\chi_{8470}(1839,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{287}{330}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{323}{330}\right)\) | \(e\left(\frac{301}{330}\right)\) |
| \(\chi_{8470}(1949,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{217}{330}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{43}{330}\right)\) | \(e\left(\frac{131}{330}\right)\) |
| \(\chi_{8470}(2019,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{193}{330}\right)\) | \(e\left(\frac{118}{165}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{277}{330}\right)\) | \(e\left(\frac{299}{330}\right)\) |
| \(\chi_{8470}(2119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{29}{330}\right)\) | \(e\left(\frac{134}{165}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{281}{330}\right)\) | \(e\left(\frac{127}{330}\right)\) |
| \(\chi_{8470}(2229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{229}{330}\right)\) | \(e\left(\frac{34}{165}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{91}{330}\right)\) | \(e\left(\frac{47}{330}\right)\) |
| \(\chi_{8470}(2329,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{311}{330}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{89}{330}\right)\) | \(e\left(\frac{133}{330}\right)\) |
| \(\chi_{8470}(2439,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{211}{330}\right)\) | \(e\left(\frac{76}{165}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{19}{330}\right)\) | \(e\left(\frac{173}{330}\right)\) |
| \(\chi_{8470}(2609,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{17}{330}\right)\) | \(e\left(\frac{107}{165}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{233}{330}\right)\) | \(e\left(\frac{211}{330}\right)\) |
| \(\chi_{8470}(2679,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{53}{330}\right)\) | \(e\left(\frac{23}{165}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{47}{330}\right)\) | \(e\left(\frac{289}{330}\right)\) |
| \(\chi_{8470}(2719,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{277}{330}\right)\) | \(e\left(\frac{142}{165}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{283}{330}\right)\) | \(e\left(\frac{41}{330}\right)\) |
| \(\chi_{8470}(2789,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{103}{330}\right)\) | \(e\left(\frac{163}{165}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{247}{330}\right)\) | \(e\left(\frac{269}{330}\right)\) |
| \(\chi_{8470}(2889,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{329}{330}\right)\) | \(e\left(\frac{149}{165}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{161}{330}\right)\) | \(e\left(\frac{7}{330}\right)\) |
| \(\chi_{8470}(2999,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{199}{330}\right)\) | \(e\left(\frac{49}{165}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{301}{330}\right)\) | \(e\left(\frac{257}{330}\right)\) |