from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6815, base_ring=CyclotomicField(644))
M = H._module
chi = DirichletCharacter(H, M([161,23,252]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,6815))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6815\) | |
| Conductor: | \(6815\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(644\) | 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: | $\Q(\zeta_{644})$ |
| Fixed field: | Number field defined by a degree 644 polynomial (not computed) |
First 31 of 264 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6815}(2,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{161}\right)\) | \(e\left(\frac{243}{322}\right)\) | \(e\left(\frac{106}{161}\right)\) | \(e\left(\frac{27}{322}\right)\) | \(e\left(\frac{129}{644}\right)\) | \(e\left(\frac{159}{161}\right)\) | \(e\left(\frac{82}{161}\right)\) | \(e\left(\frac{407}{644}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{449}{644}\right)\) |
| \(\chi_{6815}(8,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{159}{161}\right)\) | \(e\left(\frac{85}{322}\right)\) | \(e\left(\frac{157}{161}\right)\) | \(e\left(\frac{81}{322}\right)\) | \(e\left(\frac{387}{644}\right)\) | \(e\left(\frac{155}{161}\right)\) | \(e\left(\frac{85}{161}\right)\) | \(e\left(\frac{577}{644}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{59}{644}\right)\) |
| \(\chi_{6815}(18,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{135}{161}\right)\) | \(e\left(\frac{139}{322}\right)\) | \(e\left(\frac{109}{161}\right)\) | \(e\left(\frac{87}{322}\right)\) | \(e\left(\frac{523}{644}\right)\) | \(e\left(\frac{83}{161}\right)\) | \(e\left(\frac{139}{161}\right)\) | \(e\left(\frac{417}{644}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{123}{644}\right)\) |
| \(\chi_{6815}(32,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{104}{161}\right)\) | \(e\left(\frac{249}{322}\right)\) | \(e\left(\frac{47}{161}\right)\) | \(e\left(\frac{135}{322}\right)\) | \(e\left(\frac{1}{644}\right)\) | \(e\left(\frac{151}{161}\right)\) | \(e\left(\frac{88}{161}\right)\) | \(e\left(\frac{103}{644}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{313}{644}\right)\) |
| \(\chi_{6815}(68,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{148}{161}\right)\) | \(e\left(\frac{311}{322}\right)\) | \(e\left(\frac{135}{161}\right)\) | \(e\left(\frac{285}{322}\right)\) | \(e\left(\frac{503}{644}\right)\) | \(e\left(\frac{122}{161}\right)\) | \(e\left(\frac{150}{161}\right)\) | \(e\left(\frac{289}{644}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{303}{644}\right)\) |
| \(\chi_{6815}(72,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{80}{161}\right)\) | \(e\left(\frac{303}{322}\right)\) | \(e\left(\frac{160}{161}\right)\) | \(e\left(\frac{141}{322}\right)\) | \(e\left(\frac{137}{644}\right)\) | \(e\left(\frac{79}{161}\right)\) | \(e\left(\frac{142}{161}\right)\) | \(e\left(\frac{587}{644}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{377}{644}\right)\) |
| \(\chi_{6815}(143,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{161}\right)\) | \(e\left(\frac{243}{322}\right)\) | \(e\left(\frac{106}{161}\right)\) | \(e\left(\frac{27}{322}\right)\) | \(e\left(\frac{451}{644}\right)\) | \(e\left(\frac{159}{161}\right)\) | \(e\left(\frac{82}{161}\right)\) | \(e\left(\frac{85}{644}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{127}{644}\right)\) |
| \(\chi_{6815}(147,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{161}\right)\) | \(e\left(\frac{145}{322}\right)\) | \(e\left(\frac{50}{161}\right)\) | \(e\left(\frac{195}{322}\right)\) | \(e\left(\frac{73}{644}\right)\) | \(e\left(\frac{75}{161}\right)\) | \(e\left(\frac{145}{161}\right)\) | \(e\left(\frac{435}{644}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{309}{644}\right)\) |
| \(\chi_{6815}(153,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{124}{161}\right)\) | \(e\left(\frac{43}{322}\right)\) | \(e\left(\frac{87}{161}\right)\) | \(e\left(\frac{291}{322}\right)\) | \(e\left(\frac{639}{644}\right)\) | \(e\left(\frac{50}{161}\right)\) | \(e\left(\frac{43}{161}\right)\) | \(e\left(\frac{129}{644}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{367}{644}\right)\) |
| \(\chi_{6815}(177,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{161}\right)\) | \(e\left(\frac{221}{322}\right)\) | \(e\left(\frac{54}{161}\right)\) | \(e\left(\frac{275}{322}\right)\) | \(e\left(\frac{169}{644}\right)\) | \(e\left(\frac{81}{161}\right)\) | \(e\left(\frac{60}{161}\right)\) | \(e\left(\frac{19}{644}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{89}{644}\right)\) |
| \(\chi_{6815}(213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{161}\right)\) | \(e\left(\frac{73}{322}\right)\) | \(e\left(\frac{114}{161}\right)\) | \(e\left(\frac{187}{322}\right)\) | \(e\left(\frac{643}{644}\right)\) | \(e\left(\frac{10}{161}\right)\) | \(e\left(\frac{73}{161}\right)\) | \(e\left(\frac{541}{644}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{331}{644}\right)\) |
| \(\chi_{6815}(222,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{141}{161}\right)\) | \(e\left(\frac{45}{322}\right)\) | \(e\left(\frac{121}{161}\right)\) | \(e\left(\frac{5}{322}\right)\) | \(e\left(\frac{489}{644}\right)\) | \(e\left(\frac{101}{161}\right)\) | \(e\left(\frac{45}{161}\right)\) | \(e\left(\frac{135}{644}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{429}{644}\right)\) |
| \(\chi_{6815}(272,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{161}\right)\) | \(e\left(\frac{153}{322}\right)\) | \(e\left(\frac{25}{161}\right)\) | \(e\left(\frac{17}{322}\right)\) | \(e\left(\frac{117}{644}\right)\) | \(e\left(\frac{118}{161}\right)\) | \(e\left(\frac{153}{161}\right)\) | \(e\left(\frac{459}{644}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{557}{644}\right)\) |
| \(\chi_{6815}(288,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{161}\right)\) | \(e\left(\frac{145}{322}\right)\) | \(e\left(\frac{50}{161}\right)\) | \(e\left(\frac{195}{322}\right)\) | \(e\left(\frac{395}{644}\right)\) | \(e\left(\frac{75}{161}\right)\) | \(e\left(\frac{145}{161}\right)\) | \(e\left(\frac{113}{644}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{631}{644}\right)\) |
| \(\chi_{6815}(298,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{161}\right)\) | \(e\left(\frac{29}{322}\right)\) | \(e\left(\frac{10}{161}\right)\) | \(e\left(\frac{39}{322}\right)\) | \(e\left(\frac{79}{644}\right)\) | \(e\left(\frac{15}{161}\right)\) | \(e\left(\frac{29}{161}\right)\) | \(e\left(\frac{409}{644}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{255}{644}\right)\) |
| \(\chi_{6815}(363,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{161}\right)\) | \(e\left(\frac{275}{322}\right)\) | \(e\left(\frac{6}{161}\right)\) | \(e\left(\frac{281}{322}\right)\) | \(e\left(\frac{627}{644}\right)\) | \(e\left(\frac{9}{161}\right)\) | \(e\left(\frac{114}{161}\right)\) | \(e\left(\frac{181}{644}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{475}{644}\right)\) |
| \(\chi_{6815}(403,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{146}{161}\right)\) | \(e\left(\frac{235}{322}\right)\) | \(e\left(\frac{131}{161}\right)\) | \(e\left(\frac{205}{322}\right)\) | \(e\left(\frac{407}{644}\right)\) | \(e\left(\frac{116}{161}\right)\) | \(e\left(\frac{74}{161}\right)\) | \(e\left(\frac{61}{644}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{523}{644}\right)\) |
| \(\chi_{6815}(427,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{152}{161}\right)\) | \(e\left(\frac{141}{322}\right)\) | \(e\left(\frac{143}{161}\right)\) | \(e\left(\frac{123}{322}\right)\) | \(e\left(\frac{373}{644}\right)\) | \(e\left(\frac{134}{161}\right)\) | \(e\left(\frac{141}{161}\right)\) | \(e\left(\frac{423}{644}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{185}{644}\right)\) |
| \(\chi_{6815}(437,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{161}\right)\) | \(e\left(\frac{215}{322}\right)\) | \(e\left(\frac{113}{161}\right)\) | \(e\left(\frac{167}{322}\right)\) | \(e\left(\frac{297}{644}\right)\) | \(e\left(\frac{89}{161}\right)\) | \(e\left(\frac{54}{161}\right)\) | \(e\left(\frac{323}{644}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{225}{644}\right)\) |
| \(\chi_{6815}(507,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{161}\right)\) | \(e\left(\frac{107}{322}\right)\) | \(e\left(\frac{48}{161}\right)\) | \(e\left(\frac{155}{322}\right)\) | \(e\left(\frac{25}{644}\right)\) | \(e\left(\frac{72}{161}\right)\) | \(e\left(\frac{107}{161}\right)\) | \(e\left(\frac{643}{644}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{97}{644}\right)\) |
| \(\chi_{6815}(512,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{155}{161}\right)\) | \(e\left(\frac{255}{322}\right)\) | \(e\left(\frac{149}{161}\right)\) | \(e\left(\frac{243}{322}\right)\) | \(e\left(\frac{517}{644}\right)\) | \(e\left(\frac{143}{161}\right)\) | \(e\left(\frac{94}{161}\right)\) | \(e\left(\frac{443}{644}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{177}{644}\right)\) |
| \(\chi_{6815}(572,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{159}{161}\right)\) | \(e\left(\frac{85}{322}\right)\) | \(e\left(\frac{157}{161}\right)\) | \(e\left(\frac{81}{322}\right)\) | \(e\left(\frac{65}{644}\right)\) | \(e\left(\frac{155}{161}\right)\) | \(e\left(\frac{85}{161}\right)\) | \(e\left(\frac{255}{644}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{381}{644}\right)\) |
| \(\chi_{6815}(578,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{161}\right)\) | \(e\left(\frac{215}{322}\right)\) | \(e\left(\frac{113}{161}\right)\) | \(e\left(\frac{167}{322}\right)\) | \(e\left(\frac{619}{644}\right)\) | \(e\left(\frac{89}{161}\right)\) | \(e\left(\frac{54}{161}\right)\) | \(e\left(\frac{1}{644}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{547}{644}\right)\) |
| \(\chi_{6815}(582,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{158}{161}\right)\) | \(e\left(\frac{47}{322}\right)\) | \(e\left(\frac{155}{161}\right)\) | \(e\left(\frac{41}{322}\right)\) | \(e\left(\frac{17}{644}\right)\) | \(e\left(\frac{152}{161}\right)\) | \(e\left(\frac{47}{161}\right)\) | \(e\left(\frac{463}{644}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{169}{644}\right)\) |
| \(\chi_{6815}(588,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{161}\right)\) | \(e\left(\frac{309}{322}\right)\) | \(e\left(\frac{101}{161}\right)\) | \(e\left(\frac{249}{322}\right)\) | \(e\left(\frac{331}{644}\right)\) | \(e\left(\frac{71}{161}\right)\) | \(e\left(\frac{148}{161}\right)\) | \(e\left(\frac{605}{644}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{563}{644}\right)\) |
| \(\chi_{6815}(598,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{72}{161}\right)\) | \(e\left(\frac{321}{322}\right)\) | \(e\left(\frac{144}{161}\right)\) | \(e\left(\frac{143}{322}\right)\) | \(e\left(\frac{75}{644}\right)\) | \(e\left(\frac{55}{161}\right)\) | \(e\left(\frac{160}{161}\right)\) | \(e\left(\frac{641}{644}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{291}{644}\right)\) |
| \(\chi_{6815}(648,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{161}\right)\) | \(e\left(\frac{199}{322}\right)\) | \(e\left(\frac{2}{161}\right)\) | \(e\left(\frac{201}{322}\right)\) | \(e\left(\frac{531}{644}\right)\) | \(e\left(\frac{3}{161}\right)\) | \(e\left(\frac{38}{161}\right)\) | \(e\left(\frac{597}{644}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{51}{644}\right)\) |
| \(\chi_{6815}(653,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{161}\right)\) | \(e\left(\frac{163}{322}\right)\) | \(e\left(\frac{34}{161}\right)\) | \(e\left(\frac{197}{322}\right)\) | \(e\left(\frac{11}{644}\right)\) | \(e\left(\frac{51}{161}\right)\) | \(e\left(\frac{2}{161}\right)\) | \(e\left(\frac{489}{644}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{223}{644}\right)\) |
| \(\chi_{6815}(707,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{30}{161}\right)\) | \(e\left(\frac{13}{322}\right)\) | \(e\left(\frac{60}{161}\right)\) | \(e\left(\frac{73}{322}\right)\) | \(e\left(\frac{313}{644}\right)\) | \(e\left(\frac{90}{161}\right)\) | \(e\left(\frac{13}{161}\right)\) | \(e\left(\frac{39}{644}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{81}{644}\right)\) |
| \(\chi_{6815}(717,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{124}{161}\right)\) | \(e\left(\frac{43}{322}\right)\) | \(e\left(\frac{87}{161}\right)\) | \(e\left(\frac{291}{322}\right)\) | \(e\left(\frac{317}{644}\right)\) | \(e\left(\frac{50}{161}\right)\) | \(e\left(\frac{43}{161}\right)\) | \(e\left(\frac{451}{644}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{45}{644}\right)\) |
| \(\chi_{6815}(723,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{158}{161}\right)\) | \(e\left(\frac{47}{322}\right)\) | \(e\left(\frac{155}{161}\right)\) | \(e\left(\frac{41}{322}\right)\) | \(e\left(\frac{339}{644}\right)\) | \(e\left(\frac{152}{161}\right)\) | \(e\left(\frac{47}{161}\right)\) | \(e\left(\frac{141}{644}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{491}{644}\right)\) |