from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6815, base_ring=CyclotomicField(322))
M = H._module
chi = DirichletCharacter(H, M([161,23,252]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,6815))
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: | \(322\) | 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: | $\Q(\zeta_{161})$ |
| Fixed field: | Number field defined by a degree 322 polynomial (not computed) |
First 31 of 132 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6815}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{106}{161}\right)\) | \(e\left(\frac{82}{161}\right)\) | \(e\left(\frac{51}{161}\right)\) | \(e\left(\frac{27}{161}\right)\) | \(e\left(\frac{129}{322}\right)\) | \(e\left(\frac{157}{161}\right)\) | \(e\left(\frac{3}{161}\right)\) | \(e\left(\frac{85}{322}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{127}{322}\right)\) |
| \(\chi_{6815}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{82}{161}\right)\) | \(e\left(\frac{109}{161}\right)\) | \(e\left(\frac{3}{161}\right)\) | \(e\left(\frac{30}{161}\right)\) | \(e\left(\frac{197}{322}\right)\) | \(e\left(\frac{85}{161}\right)\) | \(e\left(\frac{57}{161}\right)\) | \(e\left(\frac{5}{322}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{159}{322}\right)\) |
| \(\chi_{6815}(34,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{161}\right)\) | \(e\left(\frac{34}{161}\right)\) | \(e\left(\frac{29}{161}\right)\) | \(e\left(\frac{129}{161}\right)\) | \(e\left(\frac{187}{322}\right)\) | \(e\left(\frac{124}{161}\right)\) | \(e\left(\frac{68}{161}\right)\) | \(e\left(\frac{263}{322}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{249}{322}\right)\) |
| \(\chi_{6815}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{157}{161}\right)\) | \(e\left(\frac{85}{161}\right)\) | \(e\left(\frac{153}{161}\right)\) | \(e\left(\frac{81}{161}\right)\) | \(e\left(\frac{65}{322}\right)\) | \(e\left(\frac{149}{161}\right)\) | \(e\left(\frac{9}{161}\right)\) | \(e\left(\frac{255}{322}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{59}{322}\right)\) |
| \(\chi_{6815}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{161}\right)\) | \(e\left(\frac{54}{161}\right)\) | \(e\left(\frac{65}{161}\right)\) | \(e\left(\frac{6}{161}\right)\) | \(e\left(\frac{297}{322}\right)\) | \(e\left(\frac{17}{161}\right)\) | \(e\left(\frac{108}{161}\right)\) | \(e\left(\frac{1}{322}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{225}{322}\right)\) |
| \(\chi_{6815}(209,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{161}\right)\) | \(e\left(\frac{29}{161}\right)\) | \(e\left(\frac{20}{161}\right)\) | \(e\left(\frac{39}{161}\right)\) | \(e\left(\frac{79}{322}\right)\) | \(e\left(\frac{30}{161}\right)\) | \(e\left(\frac{58}{161}\right)\) | \(e\left(\frac{87}{322}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{255}{322}\right)\) |
| \(\chi_{6815}(294,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{78}{161}\right)\) | \(e\left(\frac{33}{161}\right)\) | \(e\left(\frac{156}{161}\right)\) | \(e\left(\frac{111}{161}\right)\) | \(e\left(\frac{101}{322}\right)\) | \(e\left(\frac{73}{161}\right)\) | \(e\left(\frac{66}{161}\right)\) | \(e\left(\frac{99}{322}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{57}{322}\right)\) |
| \(\chi_{6815}(299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{161}\right)\) | \(e\left(\frac{39}{161}\right)\) | \(e\left(\frac{38}{161}\right)\) | \(e\left(\frac{58}{161}\right)\) | \(e\left(\frac{295}{322}\right)\) | \(e\left(\frac{57}{161}\right)\) | \(e\left(\frac{78}{161}\right)\) | \(e\left(\frac{117}{322}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{243}{322}\right)\) |
| \(\chi_{6815}(324,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{161}\right)\) | \(e\left(\frac{139}{161}\right)\) | \(e\left(\frac{57}{161}\right)\) | \(e\left(\frac{87}{161}\right)\) | \(e\left(\frac{201}{322}\right)\) | \(e\left(\frac{5}{161}\right)\) | \(e\left(\frac{117}{161}\right)\) | \(e\left(\frac{95}{322}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{123}{322}\right)\) |
| \(\chi_{6815}(354,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{80}{161}\right)\) | \(e\left(\frac{71}{161}\right)\) | \(e\left(\frac{160}{161}\right)\) | \(e\left(\frac{151}{161}\right)\) | \(e\left(\frac{149}{322}\right)\) | \(e\left(\frac{79}{161}\right)\) | \(e\left(\frac{142}{161}\right)\) | \(e\left(\frac{213}{322}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{269}{322}\right)\) |
| \(\chi_{6815}(439,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{120}{161}\right)\) | \(e\left(\frac{26}{161}\right)\) | \(e\left(\frac{79}{161}\right)\) | \(e\left(\frac{146}{161}\right)\) | \(e\left(\frac{143}{322}\right)\) | \(e\left(\frac{38}{161}\right)\) | \(e\left(\frac{52}{161}\right)\) | \(e\left(\frac{239}{322}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{1}{322}\right)\) |
| \(\chi_{6815}(444,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{161}\right)\) | \(e\left(\frac{144}{161}\right)\) | \(e\left(\frac{66}{161}\right)\) | \(e\left(\frac{16}{161}\right)\) | \(e\left(\frac{309}{322}\right)\) | \(e\left(\frac{99}{161}\right)\) | \(e\left(\frac{127}{161}\right)\) | \(e\left(\frac{271}{322}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{117}{322}\right)\) |
| \(\chi_{6815}(544,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{146}{161}\right)\) | \(e\left(\frac{37}{161}\right)\) | \(e\left(\frac{131}{161}\right)\) | \(e\left(\frac{22}{161}\right)\) | \(e\left(\frac{123}{322}\right)\) | \(e\left(\frac{116}{161}\right)\) | \(e\left(\frac{74}{161}\right)\) | \(e\left(\frac{111}{322}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{181}{322}\right)\) |
| \(\chi_{6815}(589,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{161}\right)\) | \(e\left(\frac{25}{161}\right)\) | \(e\left(\frac{45}{161}\right)\) | \(e\left(\frac{128}{161}\right)\) | \(e\left(\frac{57}{322}\right)\) | \(e\left(\frac{148}{161}\right)\) | \(e\left(\frac{50}{161}\right)\) | \(e\left(\frac{75}{322}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{131}{322}\right)\) |
| \(\chi_{6815}(614,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{161}\right)\) | \(e\left(\frac{20}{161}\right)\) | \(e\left(\frac{36}{161}\right)\) | \(e\left(\frac{38}{161}\right)\) | \(e\left(\frac{271}{322}\right)\) | \(e\left(\frac{54}{161}\right)\) | \(e\left(\frac{40}{161}\right)\) | \(e\left(\frac{221}{322}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{137}{322}\right)\) |
| \(\chi_{6815}(709,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{161}\right)\) | \(e\left(\frac{59}{161}\right)\) | \(e\left(\frac{74}{161}\right)\) | \(e\left(\frac{96}{161}\right)\) | \(e\left(\frac{83}{322}\right)\) | \(e\left(\frac{111}{161}\right)\) | \(e\left(\frac{118}{161}\right)\) | \(e\left(\frac{177}{322}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{219}{322}\right)\) |
| \(\chi_{6815}(729,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{161}\right)\) | \(e\left(\frac{5}{161}\right)\) | \(e\left(\frac{9}{161}\right)\) | \(e\left(\frac{90}{161}\right)\) | \(e\left(\frac{269}{322}\right)\) | \(e\left(\frac{94}{161}\right)\) | \(e\left(\frac{10}{161}\right)\) | \(e\left(\frac{15}{322}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{155}{322}\right)\) |
| \(\chi_{6815}(759,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{130}{161}\right)\) | \(e\left(\frac{55}{161}\right)\) | \(e\left(\frac{99}{161}\right)\) | \(e\left(\frac{24}{161}\right)\) | \(e\left(\frac{61}{322}\right)\) | \(e\left(\frac{68}{161}\right)\) | \(e\left(\frac{110}{161}\right)\) | \(e\left(\frac{165}{322}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{95}{322}\right)\) |
| \(\chi_{6815}(789,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{161}\right)\) | \(e\left(\frac{134}{161}\right)\) | \(e\left(\frac{48}{161}\right)\) | \(e\left(\frac{158}{161}\right)\) | \(e\left(\frac{93}{322}\right)\) | \(e\left(\frac{72}{161}\right)\) | \(e\left(\frac{107}{161}\right)\) | \(e\left(\frac{241}{322}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{129}{322}\right)\) |
| \(\chi_{6815}(854,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{161}\right)\) | \(e\left(\frac{31}{161}\right)\) | \(e\left(\frac{88}{161}\right)\) | \(e\left(\frac{75}{161}\right)\) | \(e\left(\frac{251}{322}\right)\) | \(e\left(\frac{132}{161}\right)\) | \(e\left(\frac{62}{161}\right)\) | \(e\left(\frac{93}{322}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{317}{322}\right)\) |
| \(\chi_{6815}(874,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{161}\right)\) | \(e\left(\frac{68}{161}\right)\) | \(e\left(\frac{58}{161}\right)\) | \(e\left(\frac{97}{161}\right)\) | \(e\left(\frac{213}{322}\right)\) | \(e\left(\frac{87}{161}\right)\) | \(e\left(\frac{136}{161}\right)\) | \(e\left(\frac{43}{322}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{15}{322}\right)\) |
| \(\chi_{6815}(999,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{161}\right)\) | \(e\left(\frac{10}{161}\right)\) | \(e\left(\frac{18}{161}\right)\) | \(e\left(\frac{19}{161}\right)\) | \(e\left(\frac{55}{322}\right)\) | \(e\left(\frac{27}{161}\right)\) | \(e\left(\frac{20}{161}\right)\) | \(e\left(\frac{191}{322}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{149}{322}\right)\) |
| \(\chi_{6815}(1019,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{161}\right)\) | \(e\left(\frac{159}{161}\right)\) | \(e\left(\frac{93}{161}\right)\) | \(e\left(\frac{125}{161}\right)\) | \(e\left(\frac{311}{322}\right)\) | \(e\left(\frac{59}{161}\right)\) | \(e\left(\frac{157}{161}\right)\) | \(e\left(\frac{155}{322}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{99}{322}\right)\) |
| \(\chi_{6815}(1024,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{161}\right)\) | \(e\left(\frac{88}{161}\right)\) | \(e\left(\frac{94}{161}\right)\) | \(e\left(\frac{135}{161}\right)\) | \(e\left(\frac{1}{322}\right)\) | \(e\left(\frac{141}{161}\right)\) | \(e\left(\frac{15}{161}\right)\) | \(e\left(\frac{103}{322}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{313}{322}\right)\) |
| \(\chi_{6815}(1144,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{161}\right)\) | \(e\left(\frac{3}{161}\right)\) | \(e\left(\frac{102}{161}\right)\) | \(e\left(\frac{54}{161}\right)\) | \(e\left(\frac{97}{322}\right)\) | \(e\left(\frac{153}{161}\right)\) | \(e\left(\frac{6}{161}\right)\) | \(e\left(\frac{9}{322}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{93}{322}\right)\) |
| \(\chi_{6815}(1164,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{161}\right)\) | \(e\left(\frac{145}{161}\right)\) | \(e\left(\frac{100}{161}\right)\) | \(e\left(\frac{34}{161}\right)\) | \(e\left(\frac{73}{322}\right)\) | \(e\left(\frac{150}{161}\right)\) | \(e\left(\frac{129}{161}\right)\) | \(e\left(\frac{113}{322}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{309}{322}\right)\) |
| \(\chi_{6815}(1224,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{122}{161}\right)\) | \(e\left(\frac{64}{161}\right)\) | \(e\left(\frac{83}{161}\right)\) | \(e\left(\frac{25}{161}\right)\) | \(e\left(\frac{191}{322}\right)\) | \(e\left(\frac{44}{161}\right)\) | \(e\left(\frac{128}{161}\right)\) | \(e\left(\frac{31}{322}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{213}{322}\right)\) |
| \(\chi_{6815}(1369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{94}{161}\right)\) | \(e\left(\frac{15}{161}\right)\) | \(e\left(\frac{27}{161}\right)\) | \(e\left(\frac{109}{161}\right)\) | \(e\left(\frac{163}{322}\right)\) | \(e\left(\frac{121}{161}\right)\) | \(e\left(\frac{30}{161}\right)\) | \(e\left(\frac{45}{322}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{143}{322}\right)\) |
| \(\chi_{6815}(1414,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{161}\right)\) | \(e\left(\frac{128}{161}\right)\) | \(e\left(\frac{5}{161}\right)\) | \(e\left(\frac{50}{161}\right)\) | \(e\left(\frac{221}{322}\right)\) | \(e\left(\frac{88}{161}\right)\) | \(e\left(\frac{95}{161}\right)\) | \(e\left(\frac{223}{322}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{265}{322}\right)\) |
| \(\chi_{6815}(1434,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{161}\right)\) | \(e\left(\frac{143}{161}\right)\) | \(e\left(\frac{32}{161}\right)\) | \(e\left(\frac{159}{161}\right)\) | \(e\left(\frac{223}{322}\right)\) | \(e\left(\frac{48}{161}\right)\) | \(e\left(\frac{125}{161}\right)\) | \(e\left(\frac{107}{322}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{247}{322}\right)\) |
| \(\chi_{6815}(1459,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{161}\right)\) | \(e\left(\frac{18}{161}\right)\) | \(e\left(\frac{129}{161}\right)\) | \(e\left(\frac{2}{161}\right)\) | \(e\left(\frac{99}{322}\right)\) | \(e\left(\frac{113}{161}\right)\) | \(e\left(\frac{36}{161}\right)\) | \(e\left(\frac{215}{322}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{75}{322}\right)\) |