from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4235, base_ring=CyclotomicField(660))
M = H._module
chi = DirichletCharacter(H, M([165,220,252]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,4235))
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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4235}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{197}{660}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{197}{330}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{197}{220}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{37}{132}\right)\) | \(e\left(\frac{69}{220}\right)\) | \(e\left(\frac{32}{165}\right)\) | \(e\left(\frac{193}{660}\right)\) |
| \(\chi_{4235}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{660}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{31}{330}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{31}{220}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{107}{132}\right)\) | \(e\left(\frac{127}{220}\right)\) | \(e\left(\frac{31}{165}\right)\) | \(e\left(\frac{419}{660}\right)\) |
| \(\chi_{4235}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{383}{660}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{53}{330}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{163}{220}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{171}{220}\right)\) | \(e\left(\frac{53}{165}\right)\) | \(e\left(\frac{67}{660}\right)\) |
| \(\chi_{4235}(93,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{659}{660}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{329}{330}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{219}{220}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{103}{132}\right)\) | \(e\left(\frac{3}{220}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{391}{660}\right)\) |
| \(\chi_{4235}(102,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{553}{660}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{223}{330}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{113}{220}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{65}{132}\right)\) | \(e\left(\frac{101}{220}\right)\) | \(e\left(\frac{58}{165}\right)\) | \(e\left(\frac{257}{660}\right)\) |
| \(\chi_{4235}(137,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{409}{660}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{79}{330}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{189}{220}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{113}{132}\right)\) | \(e\left(\frac{93}{220}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{461}{660}\right)\) |
| \(\chi_{4235}(158,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{307}{660}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{307}{330}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{87}{220}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{179}{220}\right)\) | \(e\left(\frac{142}{165}\right)\) | \(e\left(\frac{83}{660}\right)\) |
| \(\chi_{4235}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{660}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{191}{330}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{191}{220}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{127}{132}\right)\) | \(e\left(\frac{87}{220}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{559}{660}\right)\) |
| \(\chi_{4235}(207,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{541}{660}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{211}{330}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{101}{220}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{113}{132}\right)\) | \(e\left(\frac{137}{220}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{329}{660}\right)\) |
| \(\chi_{4235}(212,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{593}{660}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{263}{330}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{153}{220}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{37}{132}\right)\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{457}{660}\right)\) |
| \(\chi_{4235}(247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{389}{660}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{59}{330}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{169}{220}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{61}{132}\right)\) | \(e\left(\frac{153}{220}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{361}{660}\right)\) |
| \(\chi_{4235}(268,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{660}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{227}{330}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{7}{220}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{115}{132}\right)\) | \(e\left(\frac{199}{220}\right)\) | \(e\left(\frac{62}{165}\right)\) | \(e\left(\frac{343}{660}\right)\) |
| \(\chi_{4235}(312,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{217}{660}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{217}{330}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{217}{220}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{89}{132}\right)\) | \(e\left(\frac{9}{220}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{293}{660}\right)\) |
| \(\chi_{4235}(317,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{660}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{41}{330}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{41}{220}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{97}{220}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{469}{660}\right)\) |
| \(\chi_{4235}(333,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{660}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{43}{330}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{43}{220}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{91}{220}\right)\) | \(e\left(\frac{43}{165}\right)\) | \(e\left(\frac{347}{660}\right)\) |
| \(\chi_{4235}(368,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{499}{660}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{169}{330}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{59}{220}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{43}{220}\right)\) | \(e\left(\frac{4}{165}\right)\) | \(e\left(\frac{251}{660}\right)\) |
| \(\chi_{4235}(422,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{660}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{137}{330}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{137}{220}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{29}{220}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{553}{660}\right)\) |
| \(\chi_{4235}(438,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{660}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{151}{330}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{151}{220}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{23}{132}\right)\) | \(e\left(\frac{207}{220}\right)\) | \(e\left(\frac{151}{165}\right)\) | \(e\left(\frac{359}{660}\right)\) |
| \(\chi_{4235}(443,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{660}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{83}{330}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{83}{220}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{191}{220}\right)\) | \(e\left(\frac{83}{165}\right)\) | \(e\left(\frac{547}{660}\right)\) |
| \(\chi_{4235}(478,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{479}{660}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{149}{330}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{39}{220}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{149}{165}\right)\) | \(e\left(\frac{151}{660}\right)\) |
| \(\chi_{4235}(522,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{229}{660}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{229}{330}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{9}{220}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{193}{220}\right)\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{221}{660}\right)\) |
| \(\chi_{4235}(543,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{247}{660}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{247}{330}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{27}{220}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{35}{132}\right)\) | \(e\left(\frac{139}{220}\right)\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{443}{660}\right)\) |
| \(\chi_{4235}(548,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{311}{660}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{311}{330}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{91}{220}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{167}{220}\right)\) | \(e\left(\frac{146}{165}\right)\) | \(e\left(\frac{499}{660}\right)\) |
| \(\chi_{4235}(592,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{660}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{330}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{1}{220}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{217}{220}\right)\) | \(e\left(\frac{1}{165}\right)\) | \(e\left(\frac{269}{660}\right)\) |
| \(\chi_{4235}(597,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{293}{660}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{293}{330}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{73}{220}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{1}{220}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{277}{660}\right)\) |
| \(\chi_{4235}(653,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{660}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{167}{330}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{167}{220}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{91}{132}\right)\) | \(e\left(\frac{159}{220}\right)\) | \(e\left(\frac{2}{165}\right)\) | \(e\left(\frac{43}{660}\right)\) |
| \(\chi_{4235}(697,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{660}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{157}{330}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{157}{220}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{65}{132}\right)\) | \(e\left(\frac{189}{220}\right)\) | \(e\left(\frac{157}{165}\right)\) | \(e\left(\frac{653}{660}\right)\) |
| \(\chi_{4235}(702,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{660}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{161}{330}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{161}{220}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{177}{220}\right)\) | \(e\left(\frac{161}{165}\right)\) | \(e\left(\frac{409}{660}\right)\) |
| \(\chi_{4235}(718,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{403}{660}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{73}{330}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{183}{220}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{111}{220}\right)\) | \(e\left(\frac{73}{165}\right)\) | \(e\left(\frac{167}{660}\right)\) |
| \(\chi_{4235}(823,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{271}{660}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{271}{330}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{51}{220}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{67}{220}\right)\) | \(e\left(\frac{106}{165}\right)\) | \(e\left(\frac{299}{660}\right)\) |
| \(\chi_{4235}(828,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{443}{660}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{113}{330}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{3}{220}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{211}{220}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{367}{660}\right)\) |