from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3736, base_ring=CyclotomicField(466))
M = H._module
chi = DirichletCharacter(H, M([233,233,450]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,3736))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3736\) | |
| Conductor: | \(3736\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(466\) | 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: | $\Q(\zeta_{233})$ |
| Fixed field: | Number field defined by a degree 466 polynomial (not computed) |
First 31 of 232 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3736}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{128}{233}\right)\) | \(e\left(\frac{43}{466}\right)\) | \(e\left(\frac{195}{466}\right)\) | \(e\left(\frac{23}{233}\right)\) | \(e\left(\frac{37}{233}\right)\) | \(e\left(\frac{219}{466}\right)\) | \(e\left(\frac{299}{466}\right)\) | \(e\left(\frac{58}{233}\right)\) | \(e\left(\frac{166}{233}\right)\) | \(e\left(\frac{451}{466}\right)\) |
| \(\chi_{3736}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{233}\right)\) | \(e\left(\frac{129}{466}\right)\) | \(e\left(\frac{119}{466}\right)\) | \(e\left(\frac{69}{233}\right)\) | \(e\left(\frac{111}{233}\right)\) | \(e\left(\frac{191}{466}\right)\) | \(e\left(\frac{431}{466}\right)\) | \(e\left(\frac{174}{233}\right)\) | \(e\left(\frac{32}{233}\right)\) | \(e\left(\frac{421}{466}\right)\) |
| \(\chi_{3736}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{233}\right)\) | \(e\left(\frac{361}{466}\right)\) | \(e\left(\frac{445}{466}\right)\) | \(e\left(\frac{166}{233}\right)\) | \(e\left(\frac{186}{233}\right)\) | \(e\left(\frac{213}{466}\right)\) | \(e\left(\frac{61}{466}\right)\) | \(e\left(\frac{216}{233}\right)\) | \(e\left(\frac{104}{233}\right)\) | \(e\left(\frac{145}{466}\right)\) |
| \(\chi_{3736}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{186}{233}\right)\) | \(e\left(\frac{37}{466}\right)\) | \(e\left(\frac{287}{466}\right)\) | \(e\left(\frac{139}{233}\right)\) | \(e\left(\frac{21}{233}\right)\) | \(e\left(\frac{351}{466}\right)\) | \(e\left(\frac{409}{466}\right)\) | \(e\left(\frac{77}{233}\right)\) | \(e\left(\frac{132}{233}\right)\) | \(e\left(\frac{193}{466}\right)\) |
| \(\chi_{3736}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{162}{233}\right)\) | \(e\left(\frac{393}{466}\right)\) | \(e\left(\frac{265}{466}\right)\) | \(e\left(\frac{91}{233}\right)\) | \(e\left(\frac{116}{233}\right)\) | \(e\left(\frac{441}{466}\right)\) | \(e\left(\frac{251}{466}\right)\) | \(e\left(\frac{37}{233}\right)\) | \(e\left(\frac{130}{233}\right)\) | \(e\left(\frac{123}{466}\right)\) |
| \(\chi_{3736}(75,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{171}{233}\right)\) | \(e\left(\frac{143}{466}\right)\) | \(e\left(\frac{215}{466}\right)\) | \(e\left(\frac{109}{233}\right)\) | \(e\left(\frac{226}{233}\right)\) | \(e\left(\frac{349}{466}\right)\) | \(e\left(\frac{19}{466}\right)\) | \(e\left(\frac{52}{233}\right)\) | \(e\left(\frac{189}{233}\right)\) | \(e\left(\frac{91}{466}\right)\) |
| \(\chi_{3736}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{72}{233}\right)\) | \(e\left(\frac{97}{466}\right)\) | \(e\left(\frac{299}{466}\right)\) | \(e\left(\frac{144}{233}\right)\) | \(e\left(\frac{181}{233}\right)\) | \(e\left(\frac{429}{466}\right)\) | \(e\left(\frac{241}{466}\right)\) | \(e\left(\frac{120}{233}\right)\) | \(e\left(\frac{6}{233}\right)\) | \(e\left(\frac{443}{466}\right)\) |
| \(\chi_{3736}(91,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{207}{233}\right)\) | \(e\left(\frac{75}{466}\right)\) | \(e\left(\frac{15}{466}\right)\) | \(e\left(\frac{181}{233}\right)\) | \(e\left(\frac{200}{233}\right)\) | \(e\left(\frac{447}{466}\right)\) | \(e\left(\frac{23}{466}\right)\) | \(e\left(\frac{112}{233}\right)\) | \(e\left(\frac{192}{233}\right)\) | \(e\left(\frac{429}{466}\right)\) |
| \(\chi_{3736}(123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{233}\right)\) | \(e\left(\frac{251}{466}\right)\) | \(e\left(\frac{423}{466}\right)\) | \(e\left(\frac{118}{233}\right)\) | \(e\left(\frac{48}{233}\right)\) | \(e\left(\frac{303}{466}\right)\) | \(e\left(\frac{369}{466}\right)\) | \(e\left(\frac{176}{233}\right)\) | \(e\left(\frac{102}{233}\right)\) | \(e\left(\frac{75}{466}\right)\) |
| \(\chi_{3736}(139,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{140}{233}\right)\) | \(e\left(\frac{331}{466}\right)\) | \(e\left(\frac{439}{466}\right)\) | \(e\left(\frac{47}{233}\right)\) | \(e\left(\frac{106}{233}\right)\) | \(e\left(\frac{407}{466}\right)\) | \(e\left(\frac{145}{466}\right)\) | \(e\left(\frac{78}{233}\right)\) | \(e\left(\frac{167}{233}\right)\) | \(e\left(\frac{253}{466}\right)\) |
| \(\chi_{3736}(147,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{90}{233}\right)\) | \(e\left(\frac{63}{466}\right)\) | \(e\left(\frac{199}{466}\right)\) | \(e\left(\frac{180}{233}\right)\) | \(e\left(\frac{168}{233}\right)\) | \(e\left(\frac{245}{466}\right)\) | \(e\left(\frac{243}{466}\right)\) | \(e\left(\frac{150}{233}\right)\) | \(e\left(\frac{124}{233}\right)\) | \(e\left(\frac{379}{466}\right)\) |
| \(\chi_{3736}(155,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{233}\right)\) | \(e\left(\frac{231}{466}\right)\) | \(e\left(\frac{419}{466}\right)\) | \(e\left(\frac{194}{233}\right)\) | \(e\left(\frac{150}{233}\right)\) | \(e\left(\frac{277}{466}\right)\) | \(e\left(\frac{425}{466}\right)\) | \(e\left(\frac{84}{233}\right)\) | \(e\left(\frac{144}{233}\right)\) | \(e\left(\frac{147}{466}\right)\) |
| \(\chi_{3736}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{134}{233}\right)\) | \(e\left(\frac{187}{466}\right)\) | \(e\left(\frac{317}{466}\right)\) | \(e\left(\frac{35}{233}\right)\) | \(e\left(\frac{188}{233}\right)\) | \(e\left(\frac{313}{466}\right)\) | \(e\left(\frac{455}{466}\right)\) | \(e\left(\frac{68}{233}\right)\) | \(e\left(\frac{50}{233}\right)\) | \(e\left(\frac{119}{466}\right)\) |
| \(\chi_{3736}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{216}{233}\right)\) | \(e\left(\frac{291}{466}\right)\) | \(e\left(\frac{431}{466}\right)\) | \(e\left(\frac{199}{233}\right)\) | \(e\left(\frac{77}{233}\right)\) | \(e\left(\frac{355}{466}\right)\) | \(e\left(\frac{257}{466}\right)\) | \(e\left(\frac{127}{233}\right)\) | \(e\left(\frac{18}{233}\right)\) | \(e\left(\frac{397}{466}\right)\) |
| \(\chi_{3736}(227,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{233}\right)\) | \(e\left(\frac{257}{466}\right)\) | \(e\left(\frac{331}{466}\right)\) | \(e\left(\frac{2}{233}\right)\) | \(e\left(\frac{64}{233}\right)\) | \(e\left(\frac{171}{466}\right)\) | \(e\left(\frac{259}{466}\right)\) | \(e\left(\frac{157}{233}\right)\) | \(e\left(\frac{136}{233}\right)\) | \(e\left(\frac{333}{466}\right)\) |
| \(\chi_{3736}(243,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{174}{233}\right)\) | \(e\left(\frac{215}{466}\right)\) | \(e\left(\frac{43}{466}\right)\) | \(e\left(\frac{115}{233}\right)\) | \(e\left(\frac{185}{233}\right)\) | \(e\left(\frac{163}{466}\right)\) | \(e\left(\frac{97}{466}\right)\) | \(e\left(\frac{57}{233}\right)\) | \(e\left(\frac{131}{233}\right)\) | \(e\left(\frac{391}{466}\right)\) |
| \(\chi_{3736}(251,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{233}\right)\) | \(e\left(\frac{19}{466}\right)\) | \(e\left(\frac{97}{466}\right)\) | \(e\left(\frac{21}{233}\right)\) | \(e\left(\frac{206}{233}\right)\) | \(e\left(\frac{281}{466}\right)\) | \(e\left(\frac{273}{466}\right)\) | \(e\left(\frac{134}{233}\right)\) | \(e\left(\frac{30}{233}\right)\) | \(e\left(\frac{351}{466}\right)\) |
| \(\chi_{3736}(267,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{233}\right)\) | \(e\left(\frac{223}{466}\right)\) | \(e\left(\frac{231}{466}\right)\) | \(e\left(\frac{38}{233}\right)\) | \(e\left(\frac{51}{233}\right)\) | \(e\left(\frac{453}{466}\right)\) | \(e\left(\frac{261}{466}\right)\) | \(e\left(\frac{187}{233}\right)\) | \(e\left(\frac{21}{233}\right)\) | \(e\left(\frac{269}{466}\right)\) |
| \(\chi_{3736}(283,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{68}{233}\right)\) | \(e\left(\frac{1}{466}\right)\) | \(e\left(\frac{373}{466}\right)\) | \(e\left(\frac{136}{233}\right)\) | \(e\left(\frac{158}{233}\right)\) | \(e\left(\frac{211}{466}\right)\) | \(e\left(\frac{137}{466}\right)\) | \(e\left(\frac{191}{233}\right)\) | \(e\left(\frac{161}{233}\right)\) | \(e\left(\frac{43}{466}\right)\) |
| \(\chi_{3736}(291,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{233}\right)\) | \(e\left(\frac{353}{466}\right)\) | \(e\left(\frac{257}{466}\right)\) | \(e\left(\frac{10}{233}\right)\) | \(e\left(\frac{87}{233}\right)\) | \(e\left(\frac{389}{466}\right)\) | \(e\left(\frac{363}{466}\right)\) | \(e\left(\frac{86}{233}\right)\) | \(e\left(\frac{214}{233}\right)\) | \(e\left(\frac{267}{466}\right)\) |
| \(\chi_{3736}(299,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{233}\right)\) | \(e\left(\frac{409}{466}\right)\) | \(e\left(\frac{175}{466}\right)\) | \(e\left(\frac{170}{233}\right)\) | \(e\left(\frac{81}{233}\right)\) | \(e\left(\frac{89}{466}\right)\) | \(e\left(\frac{113}{466}\right)\) | \(e\left(\frac{64}{233}\right)\) | \(e\left(\frac{143}{233}\right)\) | \(e\left(\frac{345}{466}\right)\) |
| \(\chi_{3736}(331,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{34}{233}\right)\) | \(e\left(\frac{117}{466}\right)\) | \(e\left(\frac{303}{466}\right)\) | \(e\left(\frac{68}{233}\right)\) | \(e\left(\frac{79}{233}\right)\) | \(e\left(\frac{455}{466}\right)\) | \(e\left(\frac{185}{466}\right)\) | \(e\left(\frac{212}{233}\right)\) | \(e\left(\frac{197}{233}\right)\) | \(e\left(\frac{371}{466}\right)\) |
| \(\chi_{3736}(339,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{177}{233}\right)\) | \(e\left(\frac{287}{466}\right)\) | \(e\left(\frac{337}{466}\right)\) | \(e\left(\frac{121}{233}\right)\) | \(e\left(\frac{144}{233}\right)\) | \(e\left(\frac{443}{466}\right)\) | \(e\left(\frac{175}{466}\right)\) | \(e\left(\frac{62}{233}\right)\) | \(e\left(\frac{73}{233}\right)\) | \(e\left(\frac{225}{466}\right)\) |
| \(\chi_{3736}(363,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{202}{233}\right)\) | \(e\left(\frac{421}{466}\right)\) | \(e\left(\frac{457}{466}\right)\) | \(e\left(\frac{171}{233}\right)\) | \(e\left(\frac{113}{233}\right)\) | \(e\left(\frac{291}{466}\right)\) | \(e\left(\frac{359}{466}\right)\) | \(e\left(\frac{26}{233}\right)\) | \(e\left(\frac{211}{233}\right)\) | \(e\left(\frac{395}{466}\right)\) |
| \(\chi_{3736}(371,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{88}{233}\right)\) | \(e\left(\frac{15}{466}\right)\) | \(e\left(\frac{3}{466}\right)\) | \(e\left(\frac{176}{233}\right)\) | \(e\left(\frac{40}{233}\right)\) | \(e\left(\frac{369}{466}\right)\) | \(e\left(\frac{191}{466}\right)\) | \(e\left(\frac{69}{233}\right)\) | \(e\left(\frac{85}{233}\right)\) | \(e\left(\frac{179}{466}\right)\) |
| \(\chi_{3736}(387,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{106}{233}\right)\) | \(e\left(\frac{447}{466}\right)\) | \(e\left(\frac{369}{466}\right)\) | \(e\left(\frac{212}{233}\right)\) | \(e\left(\frac{27}{233}\right)\) | \(e\left(\frac{185}{466}\right)\) | \(e\left(\frac{193}{466}\right)\) | \(e\left(\frac{99}{233}\right)\) | \(e\left(\frac{203}{233}\right)\) | \(e\left(\frac{115}{466}\right)\) |
| \(\chi_{3736}(395,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{232}{233}\right)\) | \(e\left(\frac{209}{466}\right)\) | \(e\left(\frac{135}{466}\right)\) | \(e\left(\frac{231}{233}\right)\) | \(e\left(\frac{169}{233}\right)\) | \(e\left(\frac{295}{466}\right)\) | \(e\left(\frac{207}{466}\right)\) | \(e\left(\frac{76}{233}\right)\) | \(e\left(\frac{97}{233}\right)\) | \(e\left(\frac{133}{466}\right)\) |
| \(\chi_{3736}(411,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{190}{233}\right)\) | \(e\left(\frac{133}{466}\right)\) | \(e\left(\frac{213}{466}\right)\) | \(e\left(\frac{147}{233}\right)\) | \(e\left(\frac{44}{233}\right)\) | \(e\left(\frac{103}{466}\right)\) | \(e\left(\frac{47}{466}\right)\) | \(e\left(\frac{6}{233}\right)\) | \(e\left(\frac{210}{233}\right)\) | \(e\left(\frac{127}{466}\right)\) |
| \(\chi_{3736}(435,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{233}\right)\) | \(e\left(\frac{205}{466}\right)\) | \(e\left(\frac{41}{466}\right)\) | \(e\left(\frac{153}{233}\right)\) | \(e\left(\frac{3}{233}\right)\) | \(e\left(\frac{383}{466}\right)\) | \(e\left(\frac{125}{466}\right)\) | \(e\left(\frac{11}{233}\right)\) | \(e\left(\frac{152}{233}\right)\) | \(e\left(\frac{427}{466}\right)\) |
| \(\chi_{3736}(443,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{104}{233}\right)\) | \(e\left(\frac{399}{466}\right)\) | \(e\left(\frac{173}{466}\right)\) | \(e\left(\frac{208}{233}\right)\) | \(e\left(\frac{132}{233}\right)\) | \(e\left(\frac{309}{466}\right)\) | \(e\left(\frac{141}{466}\right)\) | \(e\left(\frac{18}{233}\right)\) | \(e\left(\frac{164}{233}\right)\) | \(e\left(\frac{381}{466}\right)\) |
| \(\chi_{3736}(459,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{209}{233}\right)\) | \(e\left(\frac{123}{466}\right)\) | \(e\left(\frac{211}{466}\right)\) | \(e\left(\frac{185}{233}\right)\) | \(e\left(\frac{95}{233}\right)\) | \(e\left(\frac{323}{466}\right)\) | \(e\left(\frac{75}{466}\right)\) | \(e\left(\frac{193}{233}\right)\) | \(e\left(\frac{231}{233}\right)\) | \(e\left(\frac{163}{466}\right)\) |