from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8047, base_ring=CyclotomicField(618))
M = H._module
chi = DirichletCharacter(H, M([515,294]))
chi.galois_orbit()
[g,chi] = znchar(Mod(127,8047))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8047\) | |
| Conductor: | \(8047\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(618\) | 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: | $\Q(\zeta_{309})$ |
| Fixed field: | Number field defined by a degree 618 polynomial (not computed) |
First 28 of 204 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8047}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{618}\right)\) | \(e\left(\frac{4}{309}\right)\) | \(e\left(\frac{191}{309}\right)\) | \(e\left(\frac{111}{206}\right)\) | \(e\left(\frac{199}{618}\right)\) | \(e\left(\frac{529}{618}\right)\) | \(e\left(\frac{191}{206}\right)\) | \(e\left(\frac{8}{309}\right)\) | \(e\left(\frac{262}{309}\right)\) | \(e\left(\frac{245}{618}\right)\) |
| \(\chi_{8047}(212,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{445}{618}\right)\) | \(e\left(\frac{53}{309}\right)\) | \(e\left(\frac{136}{309}\right)\) | \(e\left(\frac{3}{206}\right)\) | \(e\left(\frac{551}{618}\right)\) | \(e\left(\frac{443}{618}\right)\) | \(e\left(\frac{33}{206}\right)\) | \(e\left(\frac{106}{309}\right)\) | \(e\left(\frac{227}{309}\right)\) | \(e\left(\frac{79}{618}\right)\) |
| \(\chi_{8047}(231,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{353}{618}\right)\) | \(e\left(\frac{208}{309}\right)\) | \(e\left(\frac{44}{309}\right)\) | \(e\left(\frac{107}{206}\right)\) | \(e\left(\frac{151}{618}\right)\) | \(e\left(\frac{7}{618}\right)\) | \(e\left(\frac{147}{206}\right)\) | \(e\left(\frac{107}{309}\right)\) | \(e\left(\frac{28}{309}\right)\) | \(e\left(\frac{71}{618}\right)\) |
| \(\chi_{8047}(238,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{618}\right)\) | \(e\left(\frac{293}{309}\right)\) | \(e\left(\frac{163}{309}\right)\) | \(e\left(\frac{71}{206}\right)\) | \(e\left(\frac{131}{618}\right)\) | \(e\left(\frac{47}{618}\right)\) | \(e\left(\frac{163}{206}\right)\) | \(e\left(\frac{277}{309}\right)\) | \(e\left(\frac{188}{309}\right)\) | \(e\left(\frac{565}{618}\right)\) |
| \(\chi_{8047}(244,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{618}\right)\) | \(e\left(\frac{109}{309}\right)\) | \(e\left(\frac{29}{309}\right)\) | \(e\left(\frac{115}{206}\right)\) | \(e\left(\frac{247}{618}\right)\) | \(e\left(\frac{433}{618}\right)\) | \(e\left(\frac{29}{206}\right)\) | \(e\left(\frac{218}{309}\right)\) | \(e\left(\frac{187}{309}\right)\) | \(e\left(\frac{419}{618}\right)\) |
| \(\chi_{8047}(283,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{545}{618}\right)\) | \(e\left(\frac{301}{309}\right)\) | \(e\left(\frac{236}{309}\right)\) | \(e\left(\frac{87}{206}\right)\) | \(e\left(\frac{529}{618}\right)\) | \(e\left(\frac{487}{618}\right)\) | \(e\left(\frac{133}{206}\right)\) | \(e\left(\frac{293}{309}\right)\) | \(e\left(\frac{94}{309}\right)\) | \(e\left(\frac{437}{618}\right)\) |
| \(\chi_{8047}(342,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{618}\right)\) | \(e\left(\frac{176}{309}\right)\) | \(e\left(\frac{61}{309}\right)\) | \(e\left(\frac{43}{206}\right)\) | \(e\left(\frac{413}{618}\right)\) | \(e\left(\frac{101}{618}\right)\) | \(e\left(\frac{61}{206}\right)\) | \(e\left(\frac{43}{309}\right)\) | \(e\left(\frac{95}{309}\right)\) | \(e\left(\frac{583}{618}\right)\) |
| \(\chi_{8047}(400,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{557}{618}\right)\) | \(e\left(\frac{133}{309}\right)\) | \(e\left(\frac{248}{309}\right)\) | \(e\left(\frac{163}{206}\right)\) | \(e\left(\frac{205}{618}\right)\) | \(e\left(\frac{517}{618}\right)\) | \(e\left(\frac{145}{206}\right)\) | \(e\left(\frac{266}{309}\right)\) | \(e\left(\frac{214}{309}\right)\) | \(e\left(\frac{35}{618}\right)\) |
| \(\chi_{8047}(459,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{618}\right)\) | \(e\left(\frac{179}{309}\right)\) | \(e\left(\frac{127}{309}\right)\) | \(e\left(\frac{49}{206}\right)\) | \(e\left(\frac{485}{618}\right)\) | \(e\left(\frac{575}{618}\right)\) | \(e\left(\frac{127}{206}\right)\) | \(e\left(\frac{49}{309}\right)\) | \(e\left(\frac{137}{309}\right)\) | \(e\left(\frac{535}{618}\right)\) |
| \(\chi_{8047}(524,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{547}{618}\right)\) | \(e\left(\frac{170}{309}\right)\) | \(e\left(\frac{238}{309}\right)\) | \(e\left(\frac{31}{206}\right)\) | \(e\left(\frac{269}{618}\right)\) | \(e\left(\frac{389}{618}\right)\) | \(e\left(\frac{135}{206}\right)\) | \(e\left(\frac{31}{309}\right)\) | \(e\left(\frac{11}{309}\right)\) | \(e\left(\frac{61}{618}\right)\) |
| \(\chi_{8047}(569,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{245}{618}\right)\) | \(e\left(\frac{175}{309}\right)\) | \(e\left(\frac{245}{309}\right)\) | \(e\left(\frac{41}{206}\right)\) | \(e\left(\frac{595}{618}\right)\) | \(e\left(\frac{355}{618}\right)\) | \(e\left(\frac{39}{206}\right)\) | \(e\left(\frac{41}{309}\right)\) | \(e\left(\frac{184}{309}\right)\) | \(e\left(\frac{599}{618}\right)\) |
| \(\chi_{8047}(576,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{259}{618}\right)\) | \(e\left(\frac{185}{309}\right)\) | \(e\left(\frac{259}{309}\right)\) | \(e\left(\frac{61}{206}\right)\) | \(e\left(\frac{11}{618}\right)\) | \(e\left(\frac{287}{618}\right)\) | \(e\left(\frac{53}{206}\right)\) | \(e\left(\frac{61}{309}\right)\) | \(e\left(\frac{221}{309}\right)\) | \(e\left(\frac{439}{618}\right)\) |
| \(\chi_{8047}(602,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{211}{618}\right)\) | \(e\left(\frac{239}{309}\right)\) | \(e\left(\frac{211}{309}\right)\) | \(e\left(\frac{169}{206}\right)\) | \(e\left(\frac{71}{618}\right)\) | \(e\left(\frac{167}{618}\right)\) | \(e\left(\frac{5}{206}\right)\) | \(e\left(\frac{169}{309}\right)\) | \(e\left(\frac{50}{309}\right)\) | \(e\left(\frac{193}{618}\right)\) |
| \(\chi_{8047}(628,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{253}{618}\right)\) | \(e\left(\frac{269}{309}\right)\) | \(e\left(\frac{253}{309}\right)\) | \(e\left(\frac{23}{206}\right)\) | \(e\left(\frac{173}{618}\right)\) | \(e\left(\frac{581}{618}\right)\) | \(e\left(\frac{47}{206}\right)\) | \(e\left(\frac{229}{309}\right)\) | \(e\left(\frac{161}{309}\right)\) | \(e\left(\frac{331}{618}\right)\) |
| \(\chi_{8047}(654,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{618}\right)\) | \(e\left(\frac{8}{309}\right)\) | \(e\left(\frac{73}{309}\right)\) | \(e\left(\frac{119}{206}\right)\) | \(e\left(\frac{89}{618}\right)\) | \(e\left(\frac{131}{618}\right)\) | \(e\left(\frac{73}{206}\right)\) | \(e\left(\frac{16}{309}\right)\) | \(e\left(\frac{215}{309}\right)\) | \(e\left(\frac{181}{618}\right)\) |
| \(\chi_{8047}(706,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{481}{618}\right)\) | \(e\left(\frac{167}{309}\right)\) | \(e\left(\frac{172}{309}\right)\) | \(e\left(\frac{25}{206}\right)\) | \(e\left(\frac{197}{618}\right)\) | \(e\left(\frac{533}{618}\right)\) | \(e\left(\frac{69}{206}\right)\) | \(e\left(\frac{25}{309}\right)\) | \(e\left(\frac{278}{309}\right)\) | \(e\left(\frac{109}{618}\right)\) |
| \(\chi_{8047}(751,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{618}\right)\) | \(e\left(\frac{112}{309}\right)\) | \(e\left(\frac{95}{309}\right)\) | \(e\left(\frac{121}{206}\right)\) | \(e\left(\frac{319}{618}\right)\) | \(e\left(\frac{289}{618}\right)\) | \(e\left(\frac{95}{206}\right)\) | \(e\left(\frac{224}{309}\right)\) | \(e\left(\frac{229}{309}\right)\) | \(e\left(\frac{371}{618}\right)\) |
| \(\chi_{8047}(842,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{479}{618}\right)\) | \(e\left(\frac{298}{309}\right)\) | \(e\left(\frac{170}{309}\right)\) | \(e\left(\frac{81}{206}\right)\) | \(e\left(\frac{457}{618}\right)\) | \(e\left(\frac{13}{618}\right)\) | \(e\left(\frac{67}{206}\right)\) | \(e\left(\frac{287}{309}\right)\) | \(e\left(\frac{52}{309}\right)\) | \(e\left(\frac{485}{618}\right)\) |
| \(\chi_{8047}(1161,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{175}{618}\right)\) | \(e\left(\frac{125}{309}\right)\) | \(e\left(\frac{175}{309}\right)\) | \(e\left(\frac{147}{206}\right)\) | \(e\left(\frac{425}{618}\right)\) | \(e\left(\frac{77}{618}\right)\) | \(e\left(\frac{175}{206}\right)\) | \(e\left(\frac{250}{309}\right)\) | \(e\left(\frac{308}{309}\right)\) | \(e\left(\frac{163}{618}\right)\) |
| \(\chi_{8047}(1258,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{618}\right)\) | \(e\left(\frac{118}{309}\right)\) | \(e\left(\frac{227}{309}\right)\) | \(e\left(\frac{133}{206}\right)\) | \(e\left(\frac{463}{618}\right)\) | \(e\left(\frac{1}{618}\right)\) | \(e\left(\frac{21}{206}\right)\) | \(e\left(\frac{236}{309}\right)\) | \(e\left(\frac{4}{309}\right)\) | \(e\left(\frac{275}{618}\right)\) |
| \(\chi_{8047}(1317,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{349}{618}\right)\) | \(e\left(\frac{161}{309}\right)\) | \(e\left(\frac{40}{309}\right)\) | \(e\left(\frac{13}{206}\right)\) | \(e\left(\frac{53}{618}\right)\) | \(e\left(\frac{203}{618}\right)\) | \(e\left(\frac{143}{206}\right)\) | \(e\left(\frac{13}{309}\right)\) | \(e\left(\frac{194}{309}\right)\) | \(e\left(\frac{205}{618}\right)\) |
| \(\chi_{8047}(1330,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{499}{618}\right)\) | \(e\left(\frac{224}{309}\right)\) | \(e\left(\frac{190}{309}\right)\) | \(e\left(\frac{139}{206}\right)\) | \(e\left(\frac{329}{618}\right)\) | \(e\left(\frac{269}{618}\right)\) | \(e\left(\frac{87}{206}\right)\) | \(e\left(\frac{139}{309}\right)\) | \(e\left(\frac{149}{309}\right)\) | \(e\left(\frac{433}{618}\right)\) |
| \(\chi_{8047}(1388,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{618}\right)\) | \(e\left(\frac{52}{309}\right)\) | \(e\left(\frac{11}{309}\right)\) | \(e\left(\frac{1}{206}\right)\) | \(e\left(\frac{115}{618}\right)\) | \(e\left(\frac{79}{618}\right)\) | \(e\left(\frac{11}{206}\right)\) | \(e\left(\frac{104}{309}\right)\) | \(e\left(\frac{7}{309}\right)\) | \(e\left(\frac{95}{618}\right)\) |
| \(\chi_{8047}(1434,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{618}\right)\) | \(e\left(\frac{5}{309}\right)\) | \(e\left(\frac{7}{309}\right)\) | \(e\left(\frac{113}{206}\right)\) | \(e\left(\frac{17}{618}\right)\) | \(e\left(\frac{275}{618}\right)\) | \(e\left(\frac{7}{206}\right)\) | \(e\left(\frac{10}{309}\right)\) | \(e\left(\frac{173}{309}\right)\) | \(e\left(\frac{229}{618}\right)\) |
| \(\chi_{8047}(1447,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{618}\right)\) | \(e\left(\frac{287}{309}\right)\) | \(e\left(\frac{31}{309}\right)\) | \(e\left(\frac{59}{206}\right)\) | \(e\left(\frac{605}{618}\right)\) | \(e\left(\frac{335}{618}\right)\) | \(e\left(\frac{31}{206}\right)\) | \(e\left(\frac{265}{309}\right)\) | \(e\left(\frac{104}{309}\right)\) | \(e\left(\frac{43}{618}\right)\) |
| \(\chi_{8047}(1460,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{618}\right)\) | \(e\left(\frac{65}{309}\right)\) | \(e\left(\frac{91}{309}\right)\) | \(e\left(\frac{27}{206}\right)\) | \(e\left(\frac{221}{618}\right)\) | \(e\left(\frac{485}{618}\right)\) | \(e\left(\frac{91}{206}\right)\) | \(e\left(\frac{130}{309}\right)\) | \(e\left(\frac{86}{309}\right)\) | \(e\left(\frac{505}{618}\right)\) |
| \(\chi_{8047}(1525,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{271}{618}\right)\) | \(e\left(\frac{17}{309}\right)\) | \(e\left(\frac{271}{309}\right)\) | \(e\left(\frac{137}{206}\right)\) | \(e\left(\frac{305}{618}\right)\) | \(e\left(\frac{317}{618}\right)\) | \(e\left(\frac{65}{206}\right)\) | \(e\left(\frac{34}{309}\right)\) | \(e\left(\frac{32}{309}\right)\) | \(e\left(\frac{37}{618}\right)\) |
| \(\chi_{8047}(1531,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{401}{618}\right)\) | \(e\left(\frac{154}{309}\right)\) | \(e\left(\frac{92}{309}\right)\) | \(e\left(\frac{205}{206}\right)\) | \(e\left(\frac{91}{618}\right)\) | \(e\left(\frac{127}{618}\right)\) | \(e\left(\frac{195}{206}\right)\) | \(e\left(\frac{308}{309}\right)\) | \(e\left(\frac{199}{309}\right)\) | \(e\left(\frac{317}{618}\right)\) |