from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4235, base_ring=CyclotomicField(660))
M = H._module
chi = DirichletCharacter(H, M([165,110,294]))
chi.galois_orbit()
[g,chi] = znchar(Mod(17,4235))
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: | \(660\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{660})$ |
| Fixed field: | Number field defined by a degree 660 polynomial (not computed) |
First 31 of 160 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4235}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{660}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{19}{330}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{19}{220}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{23}{132}\right)\) | \(e\left(\frac{53}{220}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{161}{660}\right)\) |
| \(\chi_{4235}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{343}{660}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{13}{330}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{123}{220}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{181}{220}\right)\) | \(e\left(\frac{13}{165}\right)\) | \(e\left(\frac{197}{660}\right)\) |
| \(\chi_{4235}(68,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{581}{660}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{251}{330}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{141}{220}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{127}{220}\right)\) | \(e\left(\frac{86}{165}\right)\) | \(e\left(\frac{199}{660}\right)\) |
| \(\chi_{4235}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{277}{660}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{277}{330}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{57}{220}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{113}{132}\right)\) | \(e\left(\frac{159}{220}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{263}{660}\right)\) |
| \(\chi_{4235}(117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{287}{660}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{287}{330}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{67}{220}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{129}{220}\right)\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{313}{660}\right)\) |
| \(\chi_{4235}(138,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{569}{660}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{239}{330}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{129}{220}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{163}{220}\right)\) | \(e\left(\frac{74}{165}\right)\) | \(e\left(\frac{271}{660}\right)\) |
| \(\chi_{4235}(173,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{233}{660}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{233}{330}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{13}{220}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{25}{132}\right)\) | \(e\left(\frac{71}{220}\right)\) | \(e\left(\frac{68}{165}\right)\) | \(e\left(\frac{307}{660}\right)\) |
| \(\chi_{4235}(178,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{421}{660}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{91}{330}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{65}{132}\right)\) | \(e\left(\frac{167}{220}\right)\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{59}{660}\right)\) |
| \(\chi_{4235}(222,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{660}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{71}{330}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{71}{220}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{79}{132}\right)\) | \(e\left(\frac{117}{220}\right)\) | \(e\left(\frac{71}{165}\right)\) | \(e\left(\frac{289}{660}\right)\) |
| \(\chi_{4235}(227,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{367}{660}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{37}{330}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{147}{220}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{109}{220}\right)\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{53}{660}\right)\) |
| \(\chi_{4235}(248,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{589}{660}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{259}{330}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{149}{220}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{371}{660}\right)\) |
| \(\chi_{4235}(283,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{660}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{193}{330}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{193}{220}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{191}{220}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{107}{660}\right)\) |
| \(\chi_{4235}(292,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{660}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{179}{330}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{179}{220}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{13}{220}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{301}{660}\right)\) |
| \(\chi_{4235}(327,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{660}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{23}{330}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{23}{220}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{41}{220}\right)\) | \(e\left(\frac{23}{165}\right)\) | \(e\left(\frac{577}{660}\right)\) |
| \(\chi_{4235}(332,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{571}{660}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{241}{330}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{131}{220}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{157}{220}\right)\) | \(e\left(\frac{76}{165}\right)\) | \(e\left(\frac{149}{660}\right)\) |
| \(\chi_{4235}(348,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{257}{660}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{257}{330}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{37}{220}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{61}{132}\right)\) | \(e\left(\frac{219}{220}\right)\) | \(e\left(\frac{92}{165}\right)\) | \(e\left(\frac{163}{660}\right)\) |
| \(\chi_{4235}(402,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{660}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{199}{330}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{199}{220}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{173}{220}\right)\) | \(e\left(\frac{34}{165}\right)\) | \(e\left(\frac{401}{660}\right)\) |
| \(\chi_{4235}(437,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{643}{660}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{313}{330}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{203}{220}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{35}{132}\right)\) | \(e\left(\frac{161}{220}\right)\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{377}{660}\right)\) |
| \(\chi_{4235}(453,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{461}{660}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{131}{330}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{21}{220}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{37}{132}\right)\) | \(e\left(\frac{47}{220}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{259}{660}\right)\) |
| \(\chi_{4235}(458,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{337}{660}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{7}{330}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{117}{220}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{5}{132}\right)\) | \(e\left(\frac{199}{220}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{563}{660}\right)\) |
| \(\chi_{4235}(502,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{347}{660}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{17}{330}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{127}{220}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{169}{220}\right)\) | \(e\left(\frac{17}{165}\right)\) | \(e\left(\frac{613}{660}\right)\) |
| \(\chi_{4235}(523,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{660}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{89}{330}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{89}{220}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{73}{132}\right)\) | \(e\left(\frac{63}{220}\right)\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{511}{660}\right)\) |
| \(\chi_{4235}(558,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{533}{660}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{203}{330}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{93}{220}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{51}{220}\right)\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{487}{660}\right)\) |
| \(\chi_{4235}(563,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{301}{660}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{301}{330}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{81}{220}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{87}{220}\right)\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{119}{660}\right)\) |
| \(\chi_{4235}(607,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{611}{660}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{281}{330}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{171}{220}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{37}{220}\right)\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{349}{660}\right)\) |
| \(\chi_{4235}(612,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{427}{660}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{97}{330}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{207}{220}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{107}{132}\right)\) | \(e\left(\frac{149}{220}\right)\) | \(e\left(\frac{97}{165}\right)\) | \(e\left(\frac{353}{660}\right)\) |
| \(\chi_{4235}(633,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{660}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{109}{330}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{109}{220}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{3}{220}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{611}{660}\right)\) |
| \(\chi_{4235}(668,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{493}{660}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{163}{330}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{53}{220}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{171}{220}\right)\) | \(e\left(\frac{163}{165}\right)\) | \(e\left(\frac{287}{660}\right)\) |
| \(\chi_{4235}(677,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{359}{660}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{29}{330}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{139}{220}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{115}{132}\right)\) | \(e\left(\frac{133}{220}\right)\) | \(e\left(\frac{29}{165}\right)\) | \(e\left(\frac{541}{660}\right)\) |
| \(\chi_{4235}(712,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{323}{660}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{323}{330}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{127}{132}\right)\) | \(e\left(\frac{21}{220}\right)\) | \(e\left(\frac{158}{165}\right)\) | \(e\left(\frac{97}{660}\right)\) |
| \(\chi_{4235}(733,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{317}{660}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{317}{330}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{97}{220}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{39}{220}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{463}{660}\right)\) |