from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8033, base_ring=CyclotomicField(322))
M = H._module
chi = DirichletCharacter(H, M([23,308]))
chi.galois_orbit()
[g,chi] = znchar(Mod(236,8033))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8033\) | |
| Conductor: | \(8033\) | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8033}(236,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{219}{322}\right)\) | \(e\left(\frac{59}{322}\right)\) | \(e\left(\frac{58}{161}\right)\) | \(e\left(\frac{85}{161}\right)\) | \(e\left(\frac{139}{161}\right)\) | \(e\left(\frac{145}{161}\right)\) | \(e\left(\frac{13}{322}\right)\) | \(e\left(\frac{59}{161}\right)\) | \(e\left(\frac{67}{322}\right)\) | \(e\left(\frac{155}{322}\right)\) |
| \(\chi_{8033}(296,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{322}\right)\) | \(e\left(\frac{233}{322}\right)\) | \(e\left(\frac{139}{161}\right)\) | \(e\left(\frac{101}{161}\right)\) | \(e\left(\frac{25}{161}\right)\) | \(e\left(\frac{106}{161}\right)\) | \(e\left(\frac{95}{322}\right)\) | \(e\left(\frac{72}{161}\right)\) | \(e\left(\frac{19}{322}\right)\) | \(e\left(\frac{241}{322}\right)\) |
| \(\chi_{8033}(361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{322}\right)\) | \(e\left(\frac{167}{322}\right)\) | \(e\left(\frac{25}{161}\right)\) | \(e\left(\frac{156}{161}\right)\) | \(e\left(\frac{96}{161}\right)\) | \(e\left(\frac{143}{161}\right)\) | \(e\left(\frac{75}{322}\right)\) | \(e\left(\frac{6}{161}\right)\) | \(e\left(\frac{15}{322}\right)\) | \(e\left(\frac{275}{322}\right)\) |
| \(\chi_{8033}(441,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{293}{322}\right)\) | \(e\left(\frac{51}{322}\right)\) | \(e\left(\frac{132}{161}\right)\) | \(e\left(\frac{38}{161}\right)\) | \(e\left(\frac{11}{161}\right)\) | \(e\left(\frac{8}{161}\right)\) | \(e\left(\frac{235}{322}\right)\) | \(e\left(\frac{51}{161}\right)\) | \(e\left(\frac{47}{322}\right)\) | \(e\left(\frac{3}{322}\right)\) |
| \(\chi_{8033}(573,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{322}\right)\) | \(e\left(\frac{95}{322}\right)\) | \(e\left(\frac{47}{161}\right)\) | \(e\left(\frac{55}{161}\right)\) | \(e\left(\frac{71}{161}\right)\) | \(e\left(\frac{37}{161}\right)\) | \(e\left(\frac{141}{322}\right)\) | \(e\left(\frac{95}{161}\right)\) | \(e\left(\frac{157}{322}\right)\) | \(e\left(\frac{195}{322}\right)\) |
| \(\chi_{8033}(584,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{322}\right)\) | \(e\left(\frac{73}{322}\right)\) | \(e\left(\frac{9}{161}\right)\) | \(e\left(\frac{127}{161}\right)\) | \(e\left(\frac{41}{161}\right)\) | \(e\left(\frac{103}{161}\right)\) | \(e\left(\frac{27}{322}\right)\) | \(e\left(\frac{73}{161}\right)\) | \(e\left(\frac{263}{322}\right)\) | \(e\left(\frac{99}{322}\right)\) |
| \(\chi_{8033}(709,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{322}\right)\) | \(e\left(\frac{293}{322}\right)\) | \(e\left(\frac{67}{161}\right)\) | \(e\left(\frac{51}{161}\right)\) | \(e\left(\frac{19}{161}\right)\) | \(e\left(\frac{87}{161}\right)\) | \(e\left(\frac{201}{322}\right)\) | \(e\left(\frac{132}{161}\right)\) | \(e\left(\frac{169}{322}\right)\) | \(e\left(\frac{93}{322}\right)\) |
| \(\chi_{8033}(718,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{201}{322}\right)\) | \(e\left(\frac{235}{322}\right)\) | \(e\left(\frac{40}{161}\right)\) | \(e\left(\frac{153}{161}\right)\) | \(e\left(\frac{57}{161}\right)\) | \(e\left(\frac{100}{161}\right)\) | \(e\left(\frac{281}{322}\right)\) | \(e\left(\frac{74}{161}\right)\) | \(e\left(\frac{185}{322}\right)\) | \(e\left(\frac{279}{322}\right)\) |
| \(\chi_{8033}(729,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{275}{322}\right)\) | \(e\left(\frac{227}{322}\right)\) | \(e\left(\frac{114}{161}\right)\) | \(e\left(\frac{106}{161}\right)\) | \(e\left(\frac{90}{161}\right)\) | \(e\left(\frac{124}{161}\right)\) | \(e\left(\frac{181}{322}\right)\) | \(e\left(\frac{66}{161}\right)\) | \(e\left(\frac{165}{322}\right)\) | \(e\left(\frac{127}{322}\right)\) |
| \(\chi_{8033}(767,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{291}{322}\right)\) | \(e\left(\frac{321}{322}\right)\) | \(e\left(\frac{130}{161}\right)\) | \(e\left(\frac{135}{161}\right)\) | \(e\left(\frac{145}{161}\right)\) | \(e\left(\frac{3}{161}\right)\) | \(e\left(\frac{229}{322}\right)\) | \(e\left(\frac{160}{161}\right)\) | \(e\left(\frac{239}{322}\right)\) | \(e\left(\frac{303}{322}\right)\) |
| \(\chi_{8033}(818,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{307}{322}\right)\) | \(e\left(\frac{93}{322}\right)\) | \(e\left(\frac{146}{161}\right)\) | \(e\left(\frac{3}{161}\right)\) | \(e\left(\frac{39}{161}\right)\) | \(e\left(\frac{43}{161}\right)\) | \(e\left(\frac{277}{322}\right)\) | \(e\left(\frac{93}{161}\right)\) | \(e\left(\frac{313}{322}\right)\) | \(e\left(\frac{157}{322}\right)\) |
| \(\chi_{8033}(847,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{322}\right)\) | \(e\left(\frac{191}{322}\right)\) | \(e\left(\frac{125}{161}\right)\) | \(e\left(\frac{136}{161}\right)\) | \(e\left(\frac{158}{161}\right)\) | \(e\left(\frac{71}{161}\right)\) | \(e\left(\frac{53}{322}\right)\) | \(e\left(\frac{30}{161}\right)\) | \(e\left(\frac{75}{322}\right)\) | \(e\left(\frac{87}{322}\right)\) |
| \(\chi_{8033}(850,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{185}{322}\right)\) | \(e\left(\frac{141}{322}\right)\) | \(e\left(\frac{24}{161}\right)\) | \(e\left(\frac{124}{161}\right)\) | \(e\left(\frac{2}{161}\right)\) | \(e\left(\frac{60}{161}\right)\) | \(e\left(\frac{233}{322}\right)\) | \(e\left(\frac{141}{161}\right)\) | \(e\left(\frac{111}{322}\right)\) | \(e\left(\frac{103}{322}\right)\) |
| \(\chi_{8033}(883,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{322}\right)\) | \(e\left(\frac{223}{322}\right)\) | \(e\left(\frac{151}{161}\right)\) | \(e\left(\frac{2}{161}\right)\) | \(e\left(\frac{26}{161}\right)\) | \(e\left(\frac{136}{161}\right)\) | \(e\left(\frac{131}{322}\right)\) | \(e\left(\frac{62}{161}\right)\) | \(e\left(\frac{155}{322}\right)\) | \(e\left(\frac{51}{322}\right)\) |
| \(\chi_{8033}(962,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{211}{322}\right)\) | \(e\left(\frac{173}{322}\right)\) | \(e\left(\frac{50}{161}\right)\) | \(e\left(\frac{151}{161}\right)\) | \(e\left(\frac{31}{161}\right)\) | \(e\left(\frac{125}{161}\right)\) | \(e\left(\frac{311}{322}\right)\) | \(e\left(\frac{12}{161}\right)\) | \(e\left(\frac{191}{322}\right)\) | \(e\left(\frac{67}{322}\right)\) |
| \(\chi_{8033}(995,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{322}\right)\) | \(e\left(\frac{281}{322}\right)\) | \(e\left(\frac{17}{161}\right)\) | \(e\left(\frac{61}{161}\right)\) | \(e\left(\frac{149}{161}\right)\) | \(e\left(\frac{123}{161}\right)\) | \(e\left(\frac{51}{322}\right)\) | \(e\left(\frac{120}{161}\right)\) | \(e\left(\frac{139}{322}\right)\) | \(e\left(\frac{187}{322}\right)\) |
| \(\chi_{8033}(1049,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{99}{322}\right)\) | \(e\left(\frac{159}{322}\right)\) | \(e\left(\frac{99}{161}\right)\) | \(e\left(\frac{109}{161}\right)\) | \(e\left(\frac{129}{161}\right)\) | \(e\left(\frac{6}{161}\right)\) | \(e\left(\frac{297}{322}\right)\) | \(e\left(\frac{159}{161}\right)\) | \(e\left(\frac{317}{322}\right)\) | \(e\left(\frac{123}{322}\right)\) |
| \(\chi_{8033}(1095,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{215}{322}\right)\) | \(e\left(\frac{277}{322}\right)\) | \(e\left(\frac{54}{161}\right)\) | \(e\left(\frac{118}{161}\right)\) | \(e\left(\frac{85}{161}\right)\) | \(e\left(\frac{135}{161}\right)\) | \(e\left(\frac{1}{322}\right)\) | \(e\left(\frac{116}{161}\right)\) | \(e\left(\frac{129}{322}\right)\) | \(e\left(\frac{111}{322}\right)\) |
| \(\chi_{8033}(1124,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{322}\right)\) | \(e\left(\frac{53}{322}\right)\) | \(e\left(\frac{33}{161}\right)\) | \(e\left(\frac{90}{161}\right)\) | \(e\left(\frac{43}{161}\right)\) | \(e\left(\frac{2}{161}\right)\) | \(e\left(\frac{99}{322}\right)\) | \(e\left(\frac{53}{161}\right)\) | \(e\left(\frac{213}{322}\right)\) | \(e\left(\frac{41}{322}\right)\) |
| \(\chi_{8033}(1135,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{322}\right)\) | \(e\left(\frac{171}{322}\right)\) | \(e\left(\frac{149}{161}\right)\) | \(e\left(\frac{99}{161}\right)\) | \(e\left(\frac{160}{161}\right)\) | \(e\left(\frac{131}{161}\right)\) | \(e\left(\frac{125}{322}\right)\) | \(e\left(\frac{10}{161}\right)\) | \(e\left(\frac{25}{322}\right)\) | \(e\left(\frac{29}{322}\right)\) |
| \(\chi_{8033}(1309,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{322}\right)\) | \(e\left(\frac{87}{322}\right)\) | \(e\left(\frac{121}{161}\right)\) | \(e\left(\frac{8}{161}\right)\) | \(e\left(\frac{104}{161}\right)\) | \(e\left(\frac{61}{161}\right)\) | \(e\left(\frac{41}{322}\right)\) | \(e\left(\frac{87}{161}\right)\) | \(e\left(\frac{137}{322}\right)\) | \(e\left(\frac{43}{322}\right)\) |
| \(\chi_{8033}(1311,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{111}{322}\right)\) | \(e\left(\frac{149}{322}\right)\) | \(e\left(\frac{111}{161}\right)\) | \(e\left(\frac{10}{161}\right)\) | \(e\left(\frac{130}{161}\right)\) | \(e\left(\frac{36}{161}\right)\) | \(e\left(\frac{11}{322}\right)\) | \(e\left(\frac{149}{161}\right)\) | \(e\left(\frac{131}{322}\right)\) | \(e\left(\frac{255}{322}\right)\) |
| \(\chi_{8033}(1372,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{322}\right)\) | \(e\left(\frac{1}{322}\right)\) | \(e\left(\frac{31}{161}\right)\) | \(e\left(\frac{26}{161}\right)\) | \(e\left(\frac{16}{161}\right)\) | \(e\left(\frac{158}{161}\right)\) | \(e\left(\frac{93}{322}\right)\) | \(e\left(\frac{1}{161}\right)\) | \(e\left(\frac{83}{322}\right)\) | \(e\left(\frac{19}{322}\right)\) |
| \(\chi_{8033}(1401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{171}{322}\right)\) | \(e\left(\frac{99}{322}\right)\) | \(e\left(\frac{10}{161}\right)\) | \(e\left(\frac{159}{161}\right)\) | \(e\left(\frac{135}{161}\right)\) | \(e\left(\frac{25}{161}\right)\) | \(e\left(\frac{191}{322}\right)\) | \(e\left(\frac{99}{161}\right)\) | \(e\left(\frac{167}{322}\right)\) | \(e\left(\frac{271}{322}\right)\) |
| \(\chi_{8033}(1454,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{317}{322}\right)\) | \(e\left(\frac{31}{322}\right)\) | \(e\left(\frac{156}{161}\right)\) | \(e\left(\frac{1}{161}\right)\) | \(e\left(\frac{13}{161}\right)\) | \(e\left(\frac{68}{161}\right)\) | \(e\left(\frac{307}{322}\right)\) | \(e\left(\frac{31}{161}\right)\) | \(e\left(\frac{319}{322}\right)\) | \(e\left(\frac{267}{322}\right)\) |
| \(\chi_{8033}(1542,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{267}{322}\right)\) | \(e\left(\frac{19}{322}\right)\) | \(e\left(\frac{106}{161}\right)\) | \(e\left(\frac{11}{161}\right)\) | \(e\left(\frac{143}{161}\right)\) | \(e\left(\frac{104}{161}\right)\) | \(e\left(\frac{157}{322}\right)\) | \(e\left(\frac{19}{161}\right)\) | \(e\left(\frac{289}{322}\right)\) | \(e\left(\frac{39}{322}\right)\) |
| \(\chi_{8033}(1588,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{322}\right)\) | \(e\left(\frac{11}{322}\right)\) | \(e\left(\frac{19}{161}\right)\) | \(e\left(\frac{125}{161}\right)\) | \(e\left(\frac{15}{161}\right)\) | \(e\left(\frac{128}{161}\right)\) | \(e\left(\frac{57}{322}\right)\) | \(e\left(\frac{11}{161}\right)\) | \(e\left(\frac{269}{322}\right)\) | \(e\left(\frac{209}{322}\right)\) |
| \(\chi_{8033}(1658,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{281}{322}\right)\) | \(e\left(\frac{61}{322}\right)\) | \(e\left(\frac{120}{161}\right)\) | \(e\left(\frac{137}{161}\right)\) | \(e\left(\frac{10}{161}\right)\) | \(e\left(\frac{139}{161}\right)\) | \(e\left(\frac{199}{322}\right)\) | \(e\left(\frac{61}{161}\right)\) | \(e\left(\frac{233}{322}\right)\) | \(e\left(\frac{193}{322}\right)\) |
| \(\chi_{8033}(1746,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{209}{322}\right)\) | \(e\left(\frac{121}{322}\right)\) | \(e\left(\frac{48}{161}\right)\) | \(e\left(\frac{87}{161}\right)\) | \(e\left(\frac{4}{161}\right)\) | \(e\left(\frac{120}{161}\right)\) | \(e\left(\frac{305}{322}\right)\) | \(e\left(\frac{121}{161}\right)\) | \(e\left(\frac{61}{322}\right)\) | \(e\left(\frac{45}{322}\right)\) |
| \(\chi_{8033}(1831,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{177}{322}\right)\) | \(e\left(\frac{255}{322}\right)\) | \(e\left(\frac{16}{161}\right)\) | \(e\left(\frac{29}{161}\right)\) | \(e\left(\frac{55}{161}\right)\) | \(e\left(\frac{40}{161}\right)\) | \(e\left(\frac{209}{322}\right)\) | \(e\left(\frac{94}{161}\right)\) | \(e\left(\frac{235}{322}\right)\) | \(e\left(\frac{15}{322}\right)\) |
| \(\chi_{8033}(1865,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{157}{322}\right)\) | \(e\left(\frac{57}{322}\right)\) | \(e\left(\frac{157}{161}\right)\) | \(e\left(\frac{33}{161}\right)\) | \(e\left(\frac{107}{161}\right)\) | \(e\left(\frac{151}{161}\right)\) | \(e\left(\frac{149}{322}\right)\) | \(e\left(\frac{57}{161}\right)\) | \(e\left(\frac{223}{322}\right)\) | \(e\left(\frac{117}{322}\right)\) |