from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, base_ring=CyclotomicField(660))
M = H._module
chi = DirichletCharacter(H, M([165,550,588]))
chi.galois_orbit()
[g,chi] = znchar(Mod(47,8470))
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: | \(660\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 4235.dp | 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: | $\Q(\zeta_{660})$ |
| Fixed field: | Number field defined by a degree 660 polynomial (not computed) |
First 30 of 160 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8470}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{51}{220}\right)\) | \(e\left(\frac{487}{660}\right)\) | \(e\left(\frac{101}{165}\right)\) | \(e\left(\frac{103}{132}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{149}{330}\right)\) | \(e\left(\frac{221}{660}\right)\) |
| \(\chi_{8470}(103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{169}{220}\right)\) | \(e\left(\frac{613}{660}\right)\) | \(e\left(\frac{119}{165}\right)\) | \(e\left(\frac{73}{132}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{71}{330}\right)\) | \(e\left(\frac{659}{660}\right)\) |
| \(\chi_{8470}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{151}{220}\right)\) | \(e\left(\frac{467}{660}\right)\) | \(e\left(\frac{106}{165}\right)\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{109}{330}\right)\) | \(e\left(\frac{361}{660}\right)\) |
| \(\chi_{8470}(213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{189}{220}\right)\) | \(e\left(\frac{653}{660}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{151}{330}\right)\) | \(e\left(\frac{379}{660}\right)\) |
| \(\chi_{8470}(257,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{219}{220}\right)\) | \(e\left(\frac{163}{660}\right)\) | \(e\left(\frac{149}{165}\right)\) | \(e\left(\frac{67}{132}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{161}{330}\right)\) | \(e\left(\frac{509}{660}\right)\) |
| \(\chi_{8470}(313,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{13}{220}\right)\) | \(e\left(\frac{301}{660}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{107}{330}\right)\) | \(e\left(\frac{203}{660}\right)\) |
| \(\chi_{8470}(367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{19}{220}\right)\) | \(e\left(\frac{203}{660}\right)\) | \(e\left(\frac{139}{165}\right)\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{241}{330}\right)\) | \(e\left(\frac{229}{660}\right)\) |
| \(\chi_{8470}(383,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{117}{220}\right)\) | \(e\left(\frac{289}{660}\right)\) | \(e\left(\frac{2}{165}\right)\) | \(e\left(\frac{37}{132}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{83}{330}\right)\) | \(e\left(\frac{287}{660}\right)\) |
| \(\chi_{8470}(423,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{73}{220}\right)\) | \(e\left(\frac{641}{660}\right)\) | \(e\left(\frac{13}{165}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{127}{330}\right)\) | \(e\left(\frac{463}{660}\right)\) |
| \(\chi_{8470}(467,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{163}{220}\right)\) | \(e\left(\frac{271}{660}\right)\) | \(e\left(\frac{23}{165}\right)\) | \(e\left(\frac{79}{132}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{47}{330}\right)\) | \(e\left(\frac{413}{660}\right)\) |
| \(\chi_{8470}(537,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{127}{220}\right)\) | \(e\left(\frac{199}{660}\right)\) | \(e\left(\frac{107}{165}\right)\) | \(e\left(\frac{115}{132}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{233}{330}\right)\) | \(e\left(\frac{257}{660}\right)\) |
| \(\chi_{8470}(577,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{3}{220}\right)\) | \(e\left(\frac{611}{660}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{107}{132}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{67}{330}\right)\) | \(e\left(\frac{13}{660}\right)\) |
| \(\chi_{8470}(647,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{87}{220}\right)\) | \(e\left(\frac{119}{660}\right)\) | \(e\left(\frac{127}{165}\right)\) | \(e\left(\frac{23}{132}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{73}{330}\right)\) | \(e\left(\frac{157}{660}\right)\) |
| \(\chi_{8470}(663,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{61}{220}\right)\) | \(e\left(\frac{397}{660}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{299}{330}\right)\) | \(e\left(\frac{191}{660}\right)\) |
| \(\chi_{8470}(773,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{161}{220}\right)\) | \(e\left(\frac{377}{660}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{259}{330}\right)\) | \(e\left(\frac{331}{660}\right)\) |
| \(\chi_{8470}(817,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{91}{220}\right)\) | \(e\left(\frac{127}{660}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{89}{330}\right)\) | \(e\left(\frac{101}{660}\right)\) |
| \(\chi_{8470}(873,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{89}{220}\right)\) | \(e\left(\frac{13}{660}\right)\) | \(e\left(\frac{104}{165}\right)\) | \(e\left(\frac{109}{132}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{191}{330}\right)\) | \(e\left(\frac{239}{660}\right)\) |
| \(\chi_{8470}(927,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{191}{220}\right)\) | \(e\left(\frac{107}{660}\right)\) | \(e\left(\frac{31}{165}\right)\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{49}{330}\right)\) | \(e\left(\frac{241}{660}\right)\) |
| \(\chi_{8470}(983,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{109}{220}\right)\) | \(e\left(\frac{53}{660}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{89}{132}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{271}{330}\right)\) | \(e\left(\frac{619}{660}\right)\) |
| \(\chi_{8470}(1027,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{139}{220}\right)\) | \(e\left(\frac{223}{660}\right)\) | \(e\left(\frac{134}{165}\right)\) | \(e\left(\frac{103}{132}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{281}{330}\right)\) | \(e\left(\frac{89}{660}\right)\) |
| \(\chi_{8470}(1083,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{213}{220}\right)\) | \(e\left(\frac{481}{660}\right)\) | \(e\left(\frac{53}{165}\right)\) | \(e\left(\frac{73}{132}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{137}{330}\right)\) | \(e\left(\frac{263}{660}\right)\) |
| \(\chi_{8470}(1137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{159}{220}\right)\) | \(e\left(\frac{263}{660}\right)\) | \(e\left(\frac{124}{165}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{31}{330}\right)\) | \(e\left(\frac{469}{660}\right)\) |
| \(\chi_{8470}(1153,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{57}{220}\right)\) | \(e\left(\frac{169}{660}\right)\) | \(e\left(\frac{32}{165}\right)\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{173}{330}\right)\) | \(e\left(\frac{467}{660}\right)\) |
| \(\chi_{8470}(1193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{53}{220}\right)\) | \(e\left(\frac{161}{660}\right)\) | \(e\left(\frac{133}{165}\right)\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{157}{330}\right)\) | \(e\left(\frac{523}{660}\right)\) |
| \(\chi_{8470}(1263,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{220}\right)\) | \(e\left(\frac{89}{660}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{5}{132}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{13}{330}\right)\) | \(e\left(\frac{367}{660}\right)\) |
| \(\chi_{8470}(1307,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{67}{220}\right)\) | \(e\left(\frac{79}{660}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{323}{330}\right)\) | \(e\left(\frac{437}{660}\right)\) |
| \(\chi_{8470}(1347,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{203}{220}\right)\) | \(e\left(\frac{131}{660}\right)\) | \(e\left(\frac{58}{165}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{97}{330}\right)\) | \(e\left(\frac{73}{660}\right)\) |
| \(\chi_{8470}(1417,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{27}{220}\right)\) | \(e\left(\frac{659}{660}\right)\) | \(e\left(\frac{157}{165}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{163}{330}\right)\) | \(e\left(\frac{337}{660}\right)\) |
| \(\chi_{8470}(1433,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{101}{220}\right)\) | \(e\left(\frac{37}{660}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{239}{330}\right)\) | \(e\left(\frac{71}{660}\right)\) |
| \(\chi_{8470}(1543,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{17}{660}\right)\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{199}{330}\right)\) | \(e\left(\frac{211}{660}\right)\) |