from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(28900, base_ring=CyclotomicField(1360))
M = H._module
chi = DirichletCharacter(H, M([680,748,1115]))
chi.galois_orbit()
[g,chi] = znchar(Mod(23,28900))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(28900\) | |
| Conductor: | \(28900\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1360\) | 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: | $\Q(\zeta_{1360})$ |
| Fixed field: | Number field defined by a degree 1360 polynomial (not computed) |
First 27 of 512 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{28900}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{231}{1360}\right)\) | \(e\left(\frac{89}{272}\right)\) | \(e\left(\frac{231}{680}\right)\) | \(e\left(\frac{213}{1360}\right)\) | \(e\left(\frac{12}{85}\right)\) | \(e\left(\frac{597}{680}\right)\) | \(e\left(\frac{169}{340}\right)\) | \(e\left(\frac{513}{1360}\right)\) | \(e\left(\frac{693}{1360}\right)\) | \(e\left(\frac{791}{1360}\right)\) |
| \(\chi_{28900}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1259}{1360}\right)\) | \(e\left(\frac{133}{272}\right)\) | \(e\left(\frac{579}{680}\right)\) | \(e\left(\frac{737}{1360}\right)\) | \(e\left(\frac{8}{85}\right)\) | \(e\left(\frac{313}{680}\right)\) | \(e\left(\frac{141}{340}\right)\) | \(e\left(\frac{1277}{1360}\right)\) | \(e\left(\frac{1057}{1360}\right)\) | \(e\left(\frac{1179}{1360}\right)\) |
| \(\chi_{28900}(167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{273}{1360}\right)\) | \(e\left(\frac{31}{272}\right)\) | \(e\left(\frac{273}{680}\right)\) | \(e\left(\frac{499}{1360}\right)\) | \(e\left(\frac{76}{85}\right)\) | \(e\left(\frac{211}{680}\right)\) | \(e\left(\frac{107}{340}\right)\) | \(e\left(\frac{359}{1360}\right)\) | \(e\left(\frac{819}{1360}\right)\) | \(e\left(\frac{193}{1360}\right)\) |
| \(\chi_{28900}(267,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{453}{1360}\right)\) | \(e\left(\frac{171}{272}\right)\) | \(e\left(\frac{453}{680}\right)\) | \(e\left(\frac{559}{1360}\right)\) | \(e\left(\frac{71}{85}\right)\) | \(e\left(\frac{111}{680}\right)\) | \(e\left(\frac{327}{340}\right)\) | \(e\left(\frac{1059}{1360}\right)\) | \(e\left(\frac{1359}{1360}\right)\) | \(e\left(\frac{933}{1360}\right)\) |
| \(\chi_{28900}(283,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1023}{1360}\right)\) | \(e\left(\frac{161}{272}\right)\) | \(e\left(\frac{343}{680}\right)\) | \(e\left(\frac{749}{1360}\right)\) | \(e\left(\frac{41}{85}\right)\) | \(e\left(\frac{21}{680}\right)\) | \(e\left(\frac{117}{340}\right)\) | \(e\left(\frac{329}{1360}\right)\) | \(e\left(\frac{349}{1360}\right)\) | \(e\left(\frac{783}{1360}\right)\) |
| \(\chi_{28900}(363,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{439}{1360}\right)\) | \(e\left(\frac{9}{272}\right)\) | \(e\left(\frac{439}{680}\right)\) | \(e\left(\frac{917}{1360}\right)\) | \(e\left(\frac{78}{85}\right)\) | \(e\left(\frac{13}{680}\right)\) | \(e\left(\frac{121}{340}\right)\) | \(e\left(\frac{657}{1360}\right)\) | \(e\left(\frac{1317}{1360}\right)\) | \(e\left(\frac{679}{1360}\right)\) |
| \(\chi_{28900}(483,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1283}{1360}\right)\) | \(e\left(\frac{61}{272}\right)\) | \(e\left(\frac{603}{680}\right)\) | \(e\left(\frac{1289}{1360}\right)\) | \(e\left(\frac{81}{85}\right)\) | \(e\left(\frac{481}{680}\right)\) | \(e\left(\frac{57}{340}\right)\) | \(e\left(\frac{1189}{1360}\right)\) | \(e\left(\frac{1129}{1360}\right)\) | \(e\left(\frac{643}{1360}\right)\) |
| \(\chi_{28900}(547,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1337}{1360}\right)\) | \(e\left(\frac{103}{272}\right)\) | \(e\left(\frac{657}{680}\right)\) | \(e\left(\frac{491}{1360}\right)\) | \(e\left(\frac{54}{85}\right)\) | \(e\left(\frac{179}{680}\right)\) | \(e\left(\frac{123}{340}\right)\) | \(e\left(\frac{991}{1360}\right)\) | \(e\left(\frac{1291}{1360}\right)\) | \(e\left(\frac{457}{1360}\right)\) |
| \(\chi_{28900}(623,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{271}{1360}\right)\) | \(e\left(\frac{241}{272}\right)\) | \(e\left(\frac{271}{680}\right)\) | \(e\left(\frac{1133}{1360}\right)\) | \(e\left(\frac{77}{85}\right)\) | \(e\left(\frac{197}{680}\right)\) | \(e\left(\frac{29}{340}\right)\) | \(e\left(\frac{1273}{1360}\right)\) | \(e\left(\frac{813}{1360}\right)\) | \(e\left(\frac{351}{1360}\right)\) |
| \(\chi_{28900}(703,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{647}{1360}\right)\) | \(e\left(\frac{201}{272}\right)\) | \(e\left(\frac{647}{680}\right)\) | \(e\left(\frac{261}{1360}\right)\) | \(e\left(\frac{59}{85}\right)\) | \(e\left(\frac{109}{680}\right)\) | \(e\left(\frac{73}{340}\right)\) | \(e\left(\frac{801}{1360}\right)\) | \(e\left(\frac{581}{1360}\right)\) | \(e\left(\frac{567}{1360}\right)\) |
| \(\chi_{28900}(787,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1069}{1360}\right)\) | \(e\left(\frac{227}{272}\right)\) | \(e\left(\frac{389}{680}\right)\) | \(e\left(\frac{1127}{1360}\right)\) | \(e\left(\frac{18}{85}\right)\) | \(e\left(\frac{343}{680}\right)\) | \(e\left(\frac{211}{340}\right)\) | \(e\left(\frac{1067}{1360}\right)\) | \(e\left(\frac{487}{1360}\right)\) | \(e\left(\frac{1229}{1360}\right)\) |
| \(\chi_{28900}(823,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{451}{1360}\right)\) | \(e\left(\frac{109}{272}\right)\) | \(e\left(\frac{451}{680}\right)\) | \(e\left(\frac{1193}{1360}\right)\) | \(e\left(\frac{72}{85}\right)\) | \(e\left(\frac{97}{680}\right)\) | \(e\left(\frac{249}{340}\right)\) | \(e\left(\frac{613}{1360}\right)\) | \(e\left(\frac{1353}{1360}\right)\) | \(e\left(\frac{1091}{1360}\right)\) |
| \(\chi_{28900}(847,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{257}{1360}\right)\) | \(e\left(\frac{79}{272}\right)\) | \(e\left(\frac{257}{680}\right)\) | \(e\left(\frac{131}{1360}\right)\) | \(e\left(\frac{84}{85}\right)\) | \(e\left(\frac{99}{680}\right)\) | \(e\left(\frac{163}{340}\right)\) | \(e\left(\frac{871}{1360}\right)\) | \(e\left(\frac{771}{1360}\right)\) | \(e\left(\frac{97}{1360}\right)\) |
| \(\chi_{28900}(887,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1209}{1360}\right)\) | \(e\left(\frac{215}{272}\right)\) | \(e\left(\frac{529}{680}\right)\) | \(e\left(\frac{267}{1360}\right)\) | \(e\left(\frac{33}{85}\right)\) | \(e\left(\frac{643}{680}\right)\) | \(e\left(\frac{231}{340}\right)\) | \(e\left(\frac{1007}{1360}\right)\) | \(e\left(\frac{907}{1360}\right)\) | \(e\left(\frac{1049}{1360}\right)\) |
| \(\chi_{28900}(947,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{117}{1360}\right)\) | \(e\left(\frac{91}{272}\right)\) | \(e\left(\frac{117}{680}\right)\) | \(e\left(\frac{991}{1360}\right)\) | \(e\left(\frac{69}{85}\right)\) | \(e\left(\frac{479}{680}\right)\) | \(e\left(\frac{143}{340}\right)\) | \(e\left(\frac{931}{1360}\right)\) | \(e\left(\frac{351}{1360}\right)\) | \(e\left(\frac{277}{1360}\right)\) |
| \(\chi_{28900}(963,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{879}{1360}\right)\) | \(e\left(\frac{49}{272}\right)\) | \(e\left(\frac{199}{680}\right)\) | \(e\left(\frac{157}{1360}\right)\) | \(e\left(\frac{28}{85}\right)\) | \(e\left(\frac{373}{680}\right)\) | \(e\left(\frac{281}{340}\right)\) | \(e\left(\frac{857}{1360}\right)\) | \(e\left(\frac{1277}{1360}\right)\) | \(e\left(\frac{1279}{1360}\right)\) |
| \(\chi_{28900}(1127,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1101}{1360}\right)\) | \(e\left(\frac{131}{272}\right)\) | \(e\left(\frac{421}{680}\right)\) | \(e\left(\frac{503}{1360}\right)\) | \(e\left(\frac{2}{85}\right)\) | \(e\left(\frac{567}{680}\right)\) | \(e\left(\frac{99}{340}\right)\) | \(e\left(\frac{43}{1360}\right)\) | \(e\left(\frac{583}{1360}\right)\) | \(e\left(\frac{61}{1360}\right)\) |
| \(\chi_{28900}(1163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{979}{1360}\right)\) | \(e\left(\frac{157}{272}\right)\) | \(e\left(\frac{299}{680}\right)\) | \(e\left(\frac{1097}{1360}\right)\) | \(e\left(\frac{63}{85}\right)\) | \(e\left(\frac{393}{680}\right)\) | \(e\left(\frac{101}{340}\right)\) | \(e\left(\frac{37}{1360}\right)\) | \(e\left(\frac{217}{1360}\right)\) | \(e\left(\frac{179}{1360}\right)\) |
| \(\chi_{28900}(1183,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{763}{1360}\right)\) | \(e\left(\frac{261}{272}\right)\) | \(e\left(\frac{83}{680}\right)\) | \(e\left(\frac{209}{1360}\right)\) | \(e\left(\frac{1}{85}\right)\) | \(e\left(\frac{241}{680}\right)\) | \(e\left(\frac{177}{340}\right)\) | \(e\left(\frac{829}{1360}\right)\) | \(e\left(\frac{929}{1360}\right)\) | \(e\left(\frac{923}{1360}\right)\) |
| \(\chi_{28900}(1187,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{929}{1360}\right)\) | \(e\left(\frac{239}{272}\right)\) | \(e\left(\frac{249}{680}\right)\) | \(e\left(\frac{627}{1360}\right)\) | \(e\left(\frac{3}{85}\right)\) | \(e\left(\frac{43}{680}\right)\) | \(e\left(\frac{191}{340}\right)\) | \(e\left(\frac{1127}{1360}\right)\) | \(e\left(\frac{67}{1360}\right)\) | \(e\left(\frac{49}{1360}\right)\) |
| \(\chi_{28900}(1227,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1081}{1360}\right)\) | \(e\left(\frac{55}{272}\right)\) | \(e\left(\frac{401}{680}\right)\) | \(e\left(\frac{43}{1360}\right)\) | \(e\left(\frac{12}{85}\right)\) | \(e\left(\frac{427}{680}\right)\) | \(e\left(\frac{339}{340}\right)\) | \(e\left(\frac{1023}{1360}\right)\) | \(e\left(\frac{523}{1360}\right)\) | \(e\left(\frac{281}{1360}\right)\) |
| \(\chi_{28900}(1303,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{1360}\right)\) | \(e\left(\frac{129}{272}\right)\) | \(e\left(\frac{127}{680}\right)\) | \(e\left(\frac{541}{1360}\right)\) | \(e\left(\frac{64}{85}\right)\) | \(e\left(\frac{549}{680}\right)\) | \(e\left(\frac{193}{340}\right)\) | \(e\left(\frac{441}{1360}\right)\) | \(e\left(\frac{381}{1360}\right)\) | \(e\left(\frac{847}{1360}\right)\) |
| \(\chi_{28900}(1383,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1063}{1360}\right)\) | \(e\left(\frac{41}{272}\right)\) | \(e\left(\frac{383}{680}\right)\) | \(e\left(\frac{309}{1360}\right)\) | \(e\left(\frac{21}{85}\right)\) | \(e\left(\frac{301}{680}\right)\) | \(e\left(\frac{317}{340}\right)\) | \(e\left(\frac{1089}{1360}\right)\) | \(e\left(\frac{469}{1360}\right)\) | \(e\left(\frac{343}{1360}\right)\) |
| \(\chi_{28900}(1467,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1133}{1360}\right)\) | \(e\left(\frac{35}{272}\right)\) | \(e\left(\frac{453}{680}\right)\) | \(e\left(\frac{1239}{1360}\right)\) | \(e\left(\frac{71}{85}\right)\) | \(e\left(\frac{111}{680}\right)\) | \(e\left(\frac{327}{340}\right)\) | \(e\left(\frac{379}{1360}\right)\) | \(e\left(\frac{679}{1360}\right)\) | \(e\left(\frac{253}{1360}\right)\) |
| \(\chi_{28900}(1503,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{147}{1360}\right)\) | \(e\left(\frac{205}{272}\right)\) | \(e\left(\frac{147}{680}\right)\) | \(e\left(\frac{1001}{1360}\right)\) | \(e\left(\frac{54}{85}\right)\) | \(e\left(\frac{9}{680}\right)\) | \(e\left(\frac{293}{340}\right)\) | \(e\left(\frac{821}{1360}\right)\) | \(e\left(\frac{441}{1360}\right)\) | \(e\left(\frac{627}{1360}\right)\) |
| \(\chi_{28900}(1523,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1051}{1360}\right)\) | \(e\left(\frac{213}{272}\right)\) | \(e\left(\frac{371}{680}\right)\) | \(e\left(\frac{33}{1360}\right)\) | \(e\left(\frac{27}{85}\right)\) | \(e\left(\frac{217}{680}\right)\) | \(e\left(\frac{189}{340}\right)\) | \(e\left(\frac{1133}{1360}\right)\) | \(e\left(\frac{433}{1360}\right)\) | \(e\left(\frac{1291}{1360}\right)\) |
| \(\chi_{28900}(1527,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{241}{1360}\right)\) | \(e\left(\frac{127}{272}\right)\) | \(e\left(\frac{241}{680}\right)\) | \(e\left(\frac{1123}{1360}\right)\) | \(e\left(\frac{7}{85}\right)\) | \(e\left(\frac{667}{680}\right)\) | \(e\left(\frac{219}{340}\right)\) | \(e\left(\frac{23}{1360}\right)\) | \(e\left(\frac{723}{1360}\right)\) | \(e\left(\frac{1}{1360}\right)\) |