from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4235, base_ring=CyclotomicField(660))
M = H._module
chi = DirichletCharacter(H, M([165,220,6]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,4235))
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: | 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 31 of 160 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4235}(2,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{611}{660}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{281}{330}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{171}{220}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{147}{220}\right)\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{19}{660}\right)\) |
| \(\chi_{4235}(18,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{457}{660}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{127}{330}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{17}{220}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{119}{132}\right)\) | \(e\left(\frac{169}{220}\right)\) | \(e\left(\frac{127}{165}\right)\) | \(e\left(\frac{173}{660}\right)\) |
| \(\chi_{4235}(72,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{359}{660}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{29}{330}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{139}{220}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{23}{220}\right)\) | \(e\left(\frac{29}{165}\right)\) | \(e\left(\frac{211}{660}\right)\) |
| \(\chi_{4235}(107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{323}{660}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{323}{330}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{61}{132}\right)\) | \(e\left(\frac{131}{220}\right)\) | \(e\left(\frac{158}{165}\right)\) | \(e\left(\frac{427}{660}\right)\) |
| \(\chi_{4235}(123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{660}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{61}{330}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{61}{220}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{119}{132}\right)\) | \(e\left(\frac{37}{220}\right)\) | \(e\left(\frac{61}{165}\right)\) | \(e\left(\frac{569}{660}\right)\) |
| \(\chi_{4235}(128,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{317}{660}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{317}{330}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{97}{220}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{149}{220}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{133}{660}\right)\) |
| \(\chi_{4235}(172,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{547}{660}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{217}{330}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{107}{220}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{89}{132}\right)\) | \(e\left(\frac{119}{220}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{623}{660}\right)\) |
| \(\chi_{4235}(193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{469}{660}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{139}{330}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{29}{220}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{133}{220}\right)\) | \(e\left(\frac{139}{165}\right)\) | \(e\left(\frac{101}{660}\right)\) |
| \(\chi_{4235}(228,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{433}{660}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{103}{330}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{213}{220}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{21}{220}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{317}{660}\right)\) |
| \(\chi_{4235}(277,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{211}{660}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{211}{330}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{211}{220}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{113}{132}\right)\) | \(e\left(\frac{27}{220}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{659}{660}\right)\) |
| \(\chi_{4235}(303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{269}{660}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{269}{330}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{49}{220}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{79}{132}\right)\) | \(e\left(\frac{73}{220}\right)\) | \(e\left(\frac{104}{165}\right)\) | \(e\left(\frac{421}{660}\right)\) |
| \(\chi_{4235}(338,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{660}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{173}{330}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{173}{220}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{67}{132}\right)\) | \(e\left(\frac{141}{220}\right)\) | \(e\left(\frac{8}{165}\right)\) | \(e\left(\frac{337}{660}\right)\) |
| \(\chi_{4235}(347,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{660}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{79}{330}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{79}{220}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{113}{132}\right)\) | \(e\left(\frac{203}{220}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{131}{660}\right)\) |
| \(\chi_{4235}(382,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{660}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{223}{330}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{3}{220}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{65}{132}\right)\) | \(e\left(\frac{211}{220}\right)\) | \(e\left(\frac{58}{165}\right)\) | \(e\left(\frac{587}{660}\right)\) |
| \(\chi_{4235}(387,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{491}{660}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{161}{330}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{51}{220}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{67}{220}\right)\) | \(e\left(\frac{161}{165}\right)\) | \(e\left(\frac{79}{660}\right)\) |
| \(\chi_{4235}(492,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{623}{660}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{293}{330}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{183}{220}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{111}{220}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{607}{660}\right)\) |
| \(\chi_{4235}(508,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{601}{660}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{271}{330}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{161}{220}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{177}{220}\right)\) | \(e\left(\frac{106}{165}\right)\) | \(e\left(\frac{629}{660}\right)\) |
| \(\chi_{4235}(513,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{377}{660}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{47}{330}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{157}{220}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{189}{220}\right)\) | \(e\left(\frac{47}{165}\right)\) | \(e\left(\frac{433}{660}\right)\) |
| \(\chi_{4235}(557,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{607}{660}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{277}{330}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{167}{220}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{113}{132}\right)\) | \(e\left(\frac{159}{220}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{263}{660}\right)\) |
| \(\chi_{4235}(613,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{660}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{73}{330}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{73}{220}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{1}{220}\right)\) | \(e\left(\frac{73}{165}\right)\) | \(e\left(\frac{497}{660}\right)\) |
| \(\chi_{4235}(618,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{221}{660}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{221}{330}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{1}{220}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{217}{220}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{49}{660}\right)\) |
| \(\chi_{4235}(662,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{660}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{91}{330}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{91}{220}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{65}{132}\right)\) | \(e\left(\frac{167}{220}\right)\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{59}{660}\right)\) |
| \(\chi_{4235}(667,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{467}{660}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{137}{330}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{27}{220}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{139}{220}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{223}{660}\right)\) |
| \(\chi_{4235}(688,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{449}{660}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{119}{330}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{9}{220}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{193}{220}\right)\) | \(e\left(\frac{119}{165}\right)\) | \(e\left(\frac{1}{660}\right)\) |
| \(\chi_{4235}(732,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{259}{660}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{259}{330}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{39}{220}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{371}{660}\right)\) |
| \(\chi_{4235}(767,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{523}{660}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{193}{330}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{83}{220}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{191}{220}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{107}{660}\right)\) |
| \(\chi_{4235}(772,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{371}{660}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{41}{330}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{151}{220}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{207}{220}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{139}{660}\right)\) |
| \(\chi_{4235}(788,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{577}{660}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{247}{330}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{137}{220}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{35}{132}\right)\) | \(e\left(\frac{29}{220}\right)\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{113}{660}\right)\) |
| \(\chi_{4235}(842,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{660}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{59}{330}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{59}{220}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{61}{132}\right)\) | \(e\left(\frac{43}{220}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{31}{660}\right)\) |
| \(\chi_{4235}(877,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{263}{660}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{263}{330}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{43}{220}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{37}{132}\right)\) | \(e\left(\frac{91}{220}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{127}{660}\right)\) |
| \(\chi_{4235}(893,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{481}{660}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{151}{330}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{41}{220}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{23}{132}\right)\) | \(e\left(\frac{97}{220}\right)\) | \(e\left(\frac{151}{165}\right)\) | \(e\left(\frac{29}{660}\right)\) |