from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(967, base_ring=CyclotomicField(322))
M = H._module
chi = DirichletCharacter(H, M([79]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,967))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(967\) | |
| Conductor: | \(967\) | 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: | odd | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{967}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{122}{161}\right)\) | \(e\left(\frac{47}{322}\right)\) | \(e\left(\frac{83}{161}\right)\) | \(e\left(\frac{79}{322}\right)\) | \(e\left(\frac{291}{322}\right)\) | \(e\left(\frac{141}{322}\right)\) | \(e\left(\frac{44}{161}\right)\) | \(e\left(\frac{47}{161}\right)\) | \(e\left(\frac{1}{322}\right)\) | \(e\left(\frac{139}{161}\right)\) |
| \(\chi_{967}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{30}{161}\right)\) | \(e\left(\frac{1}{322}\right)\) | \(e\left(\frac{60}{161}\right)\) | \(e\left(\frac{125}{322}\right)\) | \(e\left(\frac{61}{322}\right)\) | \(e\left(\frac{3}{322}\right)\) | \(e\left(\frac{90}{161}\right)\) | \(e\left(\frac{1}{161}\right)\) | \(e\left(\frac{185}{322}\right)\) | \(e\left(\frac{116}{161}\right)\) |
| \(\chi_{967}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{81}{161}\right)\) | \(e\left(\frac{51}{322}\right)\) | \(e\left(\frac{1}{161}\right)\) | \(e\left(\frac{257}{322}\right)\) | \(e\left(\frac{213}{322}\right)\) | \(e\left(\frac{153}{322}\right)\) | \(e\left(\frac{82}{161}\right)\) | \(e\left(\frac{51}{161}\right)\) | \(e\left(\frac{97}{322}\right)\) | \(e\left(\frac{120}{161}\right)\) |
| \(\chi_{967}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{161}\right)\) | \(e\left(\frac{135}{322}\right)\) | \(e\left(\frac{50}{161}\right)\) | \(e\left(\frac{131}{322}\right)\) | \(e\left(\frac{185}{322}\right)\) | \(e\left(\frac{83}{322}\right)\) | \(e\left(\frac{75}{161}\right)\) | \(e\left(\frac{135}{161}\right)\) | \(e\left(\frac{181}{322}\right)\) | \(e\left(\frac{43}{161}\right)\) |
| \(\chi_{967}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{161}\right)\) | \(e\left(\frac{127}{322}\right)\) | \(e\left(\frac{53}{161}\right)\) | \(e\left(\frac{97}{322}\right)\) | \(e\left(\frac{19}{322}\right)\) | \(e\left(\frac{59}{322}\right)\) | \(e\left(\frac{160}{161}\right)\) | \(e\left(\frac{127}{161}\right)\) | \(e\left(\frac{311}{322}\right)\) | \(e\left(\frac{81}{161}\right)\) |
| \(\chi_{967}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{44}{161}\right)\) | \(e\left(\frac{141}{322}\right)\) | \(e\left(\frac{88}{161}\right)\) | \(e\left(\frac{237}{322}\right)\) | \(e\left(\frac{229}{322}\right)\) | \(e\left(\frac{101}{322}\right)\) | \(e\left(\frac{132}{161}\right)\) | \(e\left(\frac{141}{161}\right)\) | \(e\left(\frac{3}{322}\right)\) | \(e\left(\frac{95}{161}\right)\) |
| \(\chi_{967}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{161}\right)\) | \(e\left(\frac{159}{322}\right)\) | \(e\left(\frac{41}{161}\right)\) | \(e\left(\frac{233}{322}\right)\) | \(e\left(\frac{39}{322}\right)\) | \(e\left(\frac{155}{322}\right)\) | \(e\left(\frac{142}{161}\right)\) | \(e\left(\frac{159}{161}\right)\) | \(e\left(\frac{113}{322}\right)\) | \(e\left(\frac{90}{161}\right)\) |
| \(\chi_{967}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{90}{161}\right)\) | \(e\left(\frac{3}{322}\right)\) | \(e\left(\frac{19}{161}\right)\) | \(e\left(\frac{53}{322}\right)\) | \(e\left(\frac{183}{322}\right)\) | \(e\left(\frac{9}{322}\right)\) | \(e\left(\frac{109}{161}\right)\) | \(e\left(\frac{3}{161}\right)\) | \(e\left(\frac{233}{322}\right)\) | \(e\left(\frac{26}{161}\right)\) |
| \(\chi_{967}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{161}\right)\) | \(e\left(\frac{257}{322}\right)\) | \(e\left(\frac{125}{161}\right)\) | \(e\left(\frac{247}{322}\right)\) | \(e\left(\frac{221}{322}\right)\) | \(e\left(\frac{127}{322}\right)\) | \(e\left(\frac{107}{161}\right)\) | \(e\left(\frac{96}{161}\right)\) | \(e\left(\frac{211}{322}\right)\) | \(e\left(\frac{27}{161}\right)\) |
| \(\chi_{967}(76,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{134}{161}\right)\) | \(e\left(\frac{305}{322}\right)\) | \(e\left(\frac{107}{161}\right)\) | \(e\left(\frac{129}{322}\right)\) | \(e\left(\frac{251}{322}\right)\) | \(e\left(\frac{271}{322}\right)\) | \(e\left(\frac{80}{161}\right)\) | \(e\left(\frac{144}{161}\right)\) | \(e\left(\frac{75}{322}\right)\) | \(e\left(\frac{121}{161}\right)\) |
| \(\chi_{967}(80,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{94}{161}\right)\) | \(e\left(\frac{89}{322}\right)\) | \(e\left(\frac{27}{161}\right)\) | \(e\left(\frac{177}{322}\right)\) | \(e\left(\frac{277}{322}\right)\) | \(e\left(\frac{267}{322}\right)\) | \(e\left(\frac{121}{161}\right)\) | \(e\left(\frac{89}{161}\right)\) | \(e\left(\frac{43}{322}\right)\) | \(e\left(\frac{20}{161}\right)\) |
| \(\chi_{967}(90,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{161}\right)\) | \(e\left(\frac{95}{322}\right)\) | \(e\left(\frac{65}{161}\right)\) | \(e\left(\frac{283}{322}\right)\) | \(e\left(\frac{321}{322}\right)\) | \(e\left(\frac{285}{322}\right)\) | \(e\left(\frac{17}{161}\right)\) | \(e\left(\frac{95}{161}\right)\) | \(e\left(\frac{187}{322}\right)\) | \(e\left(\frac{72}{161}\right)\) |
| \(\chi_{967}(109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{36}{161}\right)\) | \(e\left(\frac{291}{322}\right)\) | \(e\left(\frac{72}{161}\right)\) | \(e\left(\frac{311}{322}\right)\) | \(e\left(\frac{41}{322}\right)\) | \(e\left(\frac{229}{322}\right)\) | \(e\left(\frac{108}{161}\right)\) | \(e\left(\frac{130}{161}\right)\) | \(e\left(\frac{61}{322}\right)\) | \(e\left(\frac{107}{161}\right)\) |
| \(\chi_{967}(110,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{159}{161}\right)\) | \(e\left(\frac{279}{322}\right)\) | \(e\left(\frac{157}{161}\right)\) | \(e\left(\frac{99}{322}\right)\) | \(e\left(\frac{275}{322}\right)\) | \(e\left(\frac{193}{322}\right)\) | \(e\left(\frac{155}{161}\right)\) | \(e\left(\frac{118}{161}\right)\) | \(e\left(\frac{95}{322}\right)\) | \(e\left(\frac{3}{161}\right)\) |
| \(\chi_{967}(112,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{22}{161}\right)\) | \(e\left(\frac{151}{322}\right)\) | \(e\left(\frac{44}{161}\right)\) | \(e\left(\frac{199}{322}\right)\) | \(e\left(\frac{195}{322}\right)\) | \(e\left(\frac{131}{322}\right)\) | \(e\left(\frac{66}{161}\right)\) | \(e\left(\frac{151}{161}\right)\) | \(e\left(\frac{243}{322}\right)\) | \(e\left(\frac{128}{161}\right)\) |
| \(\chi_{967}(125,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{26}{161}\right)\) | \(e\left(\frac{237}{322}\right)\) | \(e\left(\frac{52}{161}\right)\) | \(e\left(\frac{1}{322}\right)\) | \(e\left(\frac{289}{322}\right)\) | \(e\left(\frac{67}{322}\right)\) | \(e\left(\frac{78}{161}\right)\) | \(e\left(\frac{76}{161}\right)\) | \(e\left(\frac{53}{322}\right)\) | \(e\left(\frac{122}{161}\right)\) |
| \(\chi_{967}(126,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{161}\right)\) | \(e\left(\frac{157}{322}\right)\) | \(e\left(\frac{82}{161}\right)\) | \(e\left(\frac{305}{322}\right)\) | \(e\left(\frac{239}{322}\right)\) | \(e\left(\frac{149}{322}\right)\) | \(e\left(\frac{123}{161}\right)\) | \(e\left(\frac{157}{161}\right)\) | \(e\left(\frac{65}{322}\right)\) | \(e\left(\frac{19}{161}\right)\) |
| \(\chi_{967}(154,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{87}{161}\right)\) | \(e\left(\frac{19}{322}\right)\) | \(e\left(\frac{13}{161}\right)\) | \(e\left(\frac{121}{322}\right)\) | \(e\left(\frac{193}{322}\right)\) | \(e\left(\frac{57}{322}\right)\) | \(e\left(\frac{100}{161}\right)\) | \(e\left(\frac{19}{161}\right)\) | \(e\left(\frac{295}{322}\right)\) | \(e\left(\frac{111}{161}\right)\) |
| \(\chi_{967}(158,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{161}\right)\) | \(e\left(\frac{9}{322}\right)\) | \(e\left(\frac{57}{161}\right)\) | \(e\left(\frac{159}{322}\right)\) | \(e\left(\frac{227}{322}\right)\) | \(e\left(\frac{27}{322}\right)\) | \(e\left(\frac{5}{161}\right)\) | \(e\left(\frac{9}{161}\right)\) | \(e\left(\frac{55}{322}\right)\) | \(e\left(\frac{78}{161}\right)\) |
| \(\chi_{967}(167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{124}{161}\right)\) | \(e\left(\frac{251}{322}\right)\) | \(e\left(\frac{87}{161}\right)\) | \(e\left(\frac{141}{322}\right)\) | \(e\left(\frac{177}{322}\right)\) | \(e\left(\frac{109}{322}\right)\) | \(e\left(\frac{50}{161}\right)\) | \(e\left(\frac{90}{161}\right)\) | \(e\left(\frac{67}{322}\right)\) | \(e\left(\frac{136}{161}\right)\) |
| \(\chi_{967}(170,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{160}{161}\right)\) | \(e\left(\frac{59}{322}\right)\) | \(e\left(\frac{159}{161}\right)\) | \(e\left(\frac{291}{322}\right)\) | \(e\left(\frac{57}{322}\right)\) | \(e\left(\frac{177}{322}\right)\) | \(e\left(\frac{158}{161}\right)\) | \(e\left(\frac{59}{161}\right)\) | \(e\left(\frac{289}{322}\right)\) | \(e\left(\frac{82}{161}\right)\) |
| \(\chi_{967}(178,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{161}\right)\) | \(e\left(\frac{179}{322}\right)\) | \(e\left(\frac{114}{161}\right)\) | \(e\left(\frac{157}{322}\right)\) | \(e\left(\frac{293}{322}\right)\) | \(e\left(\frac{215}{322}\right)\) | \(e\left(\frac{10}{161}\right)\) | \(e\left(\frac{18}{161}\right)\) | \(e\left(\frac{271}{322}\right)\) | \(e\left(\frac{156}{161}\right)\) |
| \(\chi_{967}(184,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{161}\right)\) | \(e\left(\frac{139}{322}\right)\) | \(e\left(\frac{129}{161}\right)\) | \(e\left(\frac{309}{322}\right)\) | \(e\left(\frac{107}{322}\right)\) | \(e\left(\frac{95}{322}\right)\) | \(e\left(\frac{113}{161}\right)\) | \(e\left(\frac{139}{161}\right)\) | \(e\left(\frac{277}{322}\right)\) | \(e\left(\frac{24}{161}\right)\) |
| \(\chi_{967}(186,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{54}{161}\right)\) | \(e\left(\frac{195}{322}\right)\) | \(e\left(\frac{108}{161}\right)\) | \(e\left(\frac{225}{322}\right)\) | \(e\left(\frac{303}{322}\right)\) | \(e\left(\frac{263}{322}\right)\) | \(e\left(\frac{1}{161}\right)\) | \(e\left(\frac{34}{161}\right)\) | \(e\left(\frac{11}{322}\right)\) | \(e\left(\frac{80}{161}\right)\) |
| \(\chi_{967}(191,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{156}{161}\right)\) | \(e\left(\frac{295}{322}\right)\) | \(e\left(\frac{151}{161}\right)\) | \(e\left(\frac{167}{322}\right)\) | \(e\left(\frac{285}{322}\right)\) | \(e\left(\frac{241}{322}\right)\) | \(e\left(\frac{146}{161}\right)\) | \(e\left(\frac{134}{161}\right)\) | \(e\left(\frac{157}{322}\right)\) | \(e\left(\frac{88}{161}\right)\) |
| \(\chi_{967}(192,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{161}\right)\) | \(e\left(\frac{223}{322}\right)\) | \(e\left(\frac{17}{161}\right)\) | \(e\left(\frac{183}{322}\right)\) | \(e\left(\frac{79}{322}\right)\) | \(e\left(\frac{25}{322}\right)\) | \(e\left(\frac{106}{161}\right)\) | \(e\left(\frac{62}{161}\right)\) | \(e\left(\frac{39}{322}\right)\) | \(e\left(\frac{108}{161}\right)\) |
| \(\chi_{967}(207,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{161}\right)\) | \(e\left(\frac{145}{322}\right)\) | \(e\left(\frac{6}{161}\right)\) | \(e\left(\frac{93}{322}\right)\) | \(e\left(\frac{151}{322}\right)\) | \(e\left(\frac{113}{322}\right)\) | \(e\left(\frac{9}{161}\right)\) | \(e\left(\frac{145}{161}\right)\) | \(e\left(\frac{99}{322}\right)\) | \(e\left(\frac{76}{161}\right)\) |
| \(\chi_{967}(208,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{161}\right)\) | \(e\left(\frac{215}{322}\right)\) | \(e\left(\frac{20}{161}\right)\) | \(e\left(\frac{149}{322}\right)\) | \(e\left(\frac{235}{322}\right)\) | \(e\left(\frac{1}{322}\right)\) | \(e\left(\frac{30}{161}\right)\) | \(e\left(\frac{54}{161}\right)\) | \(e\left(\frac{169}{322}\right)\) | \(e\left(\frac{146}{161}\right)\) |
| \(\chi_{967}(213,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{161}\right)\) | \(e\left(\frac{125}{322}\right)\) | \(e\left(\frac{94}{161}\right)\) | \(e\left(\frac{169}{322}\right)\) | \(e\left(\frac{219}{322}\right)\) | \(e\left(\frac{53}{322}\right)\) | \(e\left(\frac{141}{161}\right)\) | \(e\left(\frac{125}{161}\right)\) | \(e\left(\frac{263}{322}\right)\) | \(e\left(\frac{10}{161}\right)\) |
| \(\chi_{967}(214,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{142}{161}\right)\) | \(e\left(\frac{155}{322}\right)\) | \(e\left(\frac{123}{161}\right)\) | \(e\left(\frac{55}{322}\right)\) | \(e\left(\frac{117}{322}\right)\) | \(e\left(\frac{143}{322}\right)\) | \(e\left(\frac{104}{161}\right)\) | \(e\left(\frac{155}{161}\right)\) | \(e\left(\frac{17}{322}\right)\) | \(e\left(\frac{109}{161}\right)\) |
| \(\chi_{967}(216,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{108}{161}\right)\) | \(e\left(\frac{229}{322}\right)\) | \(e\left(\frac{55}{161}\right)\) | \(e\left(\frac{289}{322}\right)\) | \(e\left(\frac{123}{322}\right)\) | \(e\left(\frac{43}{322}\right)\) | \(e\left(\frac{2}{161}\right)\) | \(e\left(\frac{68}{161}\right)\) | \(e\left(\frac{183}{322}\right)\) | \(e\left(\frac{160}{161}\right)\) |