from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10015, base_ring=CyclotomicField(4004))
M = H._module
chi = DirichletCharacter(H, M([3003,692]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,10015))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(10015\) | |
| Conductor: | \(10015\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(4004\) | 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: | $\Q(\zeta_{4004})$ |
| Fixed field: | Number field defined by a degree 4004 polynomial (not computed) |
First 31 of 1440 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{10015}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{469}{572}\right)\) | \(e\left(\frac{193}{4004}\right)\) | \(e\left(\frac{183}{286}\right)\) | \(e\left(\frac{79}{91}\right)\) | \(e\left(\frac{31}{4004}\right)\) | \(e\left(\frac{263}{572}\right)\) | \(e\left(\frac{193}{2002}\right)\) | \(e\left(\frac{67}{143}\right)\) | \(e\left(\frac{2755}{4004}\right)\) | \(e\left(\frac{1817}{4004}\right)\) |
| \(\chi_{10015}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{572}\right)\) | \(e\left(\frac{2755}{4004}\right)\) | \(e\left(\frac{59}{286}\right)\) | \(e\left(\frac{72}{91}\right)\) | \(e\left(\frac{3181}{4004}\right)\) | \(e\left(\frac{177}{572}\right)\) | \(e\left(\frac{753}{2002}\right)\) | \(e\left(\frac{131}{143}\right)\) | \(e\left(\frac{3581}{4004}\right)\) | \(e\left(\frac{1747}{4004}\right)\) |
| \(\chi_{10015}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{281}{572}\right)\) | \(e\left(\frac{1817}{4004}\right)\) | \(e\left(\frac{281}{286}\right)\) | \(e\left(\frac{86}{91}\right)\) | \(e\left(\frac{2159}{4004}\right)\) | \(e\left(\frac{271}{572}\right)\) | \(e\left(\frac{1817}{2002}\right)\) | \(e\left(\frac{81}{143}\right)\) | \(e\left(\frac{1747}{4004}\right)\) | \(e\left(\frac{613}{4004}\right)\) |
| \(\chi_{10015}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{263}{572}\right)\) | \(e\left(\frac{579}{4004}\right)\) | \(e\left(\frac{263}{286}\right)\) | \(e\left(\frac{55}{91}\right)\) | \(e\left(\frac{93}{4004}\right)\) | \(e\left(\frac{217}{572}\right)\) | \(e\left(\frac{579}{2002}\right)\) | \(e\left(\frac{58}{143}\right)\) | \(e\left(\frac{257}{4004}\right)\) | \(e\left(\frac{1447}{4004}\right)\) |
| \(\chi_{10015}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{87}{572}\right)\) | \(e\left(\frac{2075}{4004}\right)\) | \(e\left(\frac{87}{286}\right)\) | \(e\left(\frac{61}{91}\right)\) | \(e\left(\frac{3217}{4004}\right)\) | \(e\left(\frac{261}{572}\right)\) | \(e\left(\frac{73}{2002}\right)\) | \(e\left(\frac{135}{143}\right)\) | \(e\left(\frac{3293}{4004}\right)\) | \(e\left(\frac{1403}{4004}\right)\) |
| \(\chi_{10015}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{443}{572}\right)\) | \(e\left(\frac{375}{4004}\right)\) | \(e\left(\frac{157}{286}\right)\) | \(e\left(\frac{79}{91}\right)\) | \(e\left(\frac{1305}{4004}\right)\) | \(e\left(\frac{185}{572}\right)\) | \(e\left(\frac{375}{2002}\right)\) | \(e\left(\frac{2}{143}\right)\) | \(e\left(\frac{2573}{4004}\right)\) | \(e\left(\frac{543}{4004}\right)\) |
| \(\chi_{10015}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{572}\right)\) | \(e\left(\frac{857}{4004}\right)\) | \(e\left(\frac{85}{286}\right)\) | \(e\left(\frac{33}{91}\right)\) | \(e\left(\frac{3623}{4004}\right)\) | \(e\left(\frac{255}{572}\right)\) | \(e\left(\frac{857}{2002}\right)\) | \(e\left(\frac{53}{143}\right)\) | \(e\left(\frac{2047}{4004}\right)\) | \(e\left(\frac{1305}{4004}\right)\) |
| \(\chi_{10015}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{335}{572}\right)\) | \(e\left(\frac{1527}{4004}\right)\) | \(e\left(\frac{49}{286}\right)\) | \(e\left(\frac{88}{91}\right)\) | \(e\left(\frac{349}{4004}\right)\) | \(e\left(\frac{433}{572}\right)\) | \(e\left(\frac{1527}{2002}\right)\) | \(e\left(\frac{7}{143}\right)\) | \(e\left(\frac{2213}{4004}\right)\) | \(e\left(\frac{2115}{4004}\right)\) |
| \(\chi_{10015}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{285}{572}\right)\) | \(e\left(\frac{821}{4004}\right)\) | \(e\left(\frac{285}{286}\right)\) | \(e\left(\frac{64}{91}\right)\) | \(e\left(\frac{775}{4004}\right)\) | \(e\left(\frac{283}{572}\right)\) | \(e\left(\frac{821}{2002}\right)\) | \(e\left(\frac{102}{143}\right)\) | \(e\left(\frac{807}{4004}\right)\) | \(e\left(\frac{1381}{4004}\right)\) |
| \(\chi_{10015}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{572}\right)\) | \(e\left(\frac{1907}{4004}\right)\) | \(e\left(\frac{67}{286}\right)\) | \(e\left(\frac{54}{91}\right)\) | \(e\left(\frac{3273}{4004}\right)\) | \(e\left(\frac{201}{572}\right)\) | \(e\left(\frac{1907}{2002}\right)\) | \(e\left(\frac{30}{143}\right)\) | \(e\left(\frac{2845}{4004}\right)\) | \(e\left(\frac{423}{4004}\right)\) |
| \(\chi_{10015}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{467}{572}\right)\) | \(e\left(\frac{691}{4004}\right)\) | \(e\left(\frac{181}{286}\right)\) | \(e\left(\frac{90}{91}\right)\) | \(e\left(\frac{2725}{4004}\right)\) | \(e\left(\frac{257}{572}\right)\) | \(e\left(\frac{691}{2002}\right)\) | \(e\left(\frac{128}{143}\right)\) | \(e\left(\frac{3225}{4004}\right)\) | \(e\left(\frac{3435}{4004}\right)\) |
| \(\chi_{10015}(107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{335}{572}\right)\) | \(e\left(\frac{3243}{4004}\right)\) | \(e\left(\frac{49}{286}\right)\) | \(e\left(\frac{36}{91}\right)\) | \(e\left(\frac{2637}{4004}\right)\) | \(e\left(\frac{433}{572}\right)\) | \(e\left(\frac{1241}{2002}\right)\) | \(e\left(\frac{7}{143}\right)\) | \(e\left(\frac{3929}{4004}\right)\) | \(e\left(\frac{3831}{4004}\right)\) |
| \(\chi_{10015}(108,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{425}{572}\right)\) | \(e\left(\frac{3141}{4004}\right)\) | \(e\left(\frac{139}{286}\right)\) | \(e\left(\frac{48}{91}\right)\) | \(e\left(\frac{3243}{4004}\right)\) | \(e\left(\frac{131}{572}\right)\) | \(e\left(\frac{1139}{2002}\right)\) | \(e\left(\frac{122}{143}\right)\) | \(e\left(\frac{1083}{4004}\right)\) | \(e\left(\frac{1377}{4004}\right)\) |
| \(\chi_{10015}(117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{75}{572}\right)\) | \(e\left(\frac{2203}{4004}\right)\) | \(e\left(\frac{75}{286}\right)\) | \(e\left(\frac{62}{91}\right)\) | \(e\left(\frac{2221}{4004}\right)\) | \(e\left(\frac{225}{572}\right)\) | \(e\left(\frac{201}{2002}\right)\) | \(e\left(\frac{72}{143}\right)\) | \(e\left(\frac{3253}{4004}\right)\) | \(e\left(\frac{243}{4004}\right)\) |
| \(\chi_{10015}(122,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{519}{572}\right)\) | \(e\left(\frac{3187}{4004}\right)\) | \(e\left(\frac{233}{286}\right)\) | \(e\left(\frac{64}{91}\right)\) | \(e\left(\frac{1321}{4004}\right)\) | \(e\left(\frac{413}{572}\right)\) | \(e\left(\frac{1185}{2002}\right)\) | \(e\left(\frac{115}{143}\right)\) | \(e\left(\frac{2445}{4004}\right)\) | \(e\left(\frac{835}{4004}\right)\) |
| \(\chi_{10015}(138,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{301}{572}\right)\) | \(e\left(\frac{1985}{4004}\right)\) | \(e\left(\frac{15}{286}\right)\) | \(e\left(\frac{2}{91}\right)\) | \(e\left(\frac{2103}{4004}\right)\) | \(e\left(\frac{331}{572}\right)\) | \(e\left(\frac{1985}{2002}\right)\) | \(e\left(\frac{43}{143}\right)\) | \(e\left(\frac{2195}{4004}\right)\) | \(e\left(\frac{1593}{4004}\right)\) |
| \(\chi_{10015}(147,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{347}{572}\right)\) | \(e\left(\frac{255}{4004}\right)\) | \(e\left(\frac{61}{286}\right)\) | \(e\left(\frac{61}{91}\right)\) | \(e\left(\frac{2489}{4004}\right)\) | \(e\left(\frac{469}{572}\right)\) | \(e\left(\frac{255}{2002}\right)\) | \(e\left(\frac{70}{143}\right)\) | \(e\left(\frac{1109}{4004}\right)\) | \(e\left(\frac{2131}{4004}\right)\) |
| \(\chi_{10015}(167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{572}\right)\) | \(e\left(\frac{3251}{4004}\right)\) | \(e\left(\frac{227}{286}\right)\) | \(e\left(\frac{19}{91}\right)\) | \(e\left(\frac{2825}{4004}\right)\) | \(e\left(\frac{109}{572}\right)\) | \(e\left(\frac{1249}{2002}\right)\) | \(e\left(\frac{12}{143}\right)\) | \(e\left(\frac{2425}{4004}\right)\) | \(e\left(\frac{255}{4004}\right)\) |
| \(\chi_{10015}(173,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{369}{572}\right)\) | \(e\left(\frac{1069}{4004}\right)\) | \(e\left(\frac{83}{286}\right)\) | \(e\left(\frac{83}{91}\right)\) | \(e\left(\frac{2599}{4004}\right)\) | \(e\left(\frac{535}{572}\right)\) | \(e\left(\frac{1069}{2002}\right)\) | \(e\left(\frac{114}{143}\right)\) | \(e\left(\frac{2231}{4004}\right)\) | \(e\left(\frac{2637}{4004}\right)\) |
| \(\chi_{10015}(177,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{483}{572}\right)\) | \(e\left(\frac{2999}{4004}\right)\) | \(e\left(\frac{197}{286}\right)\) | \(e\left(\frac{54}{91}\right)\) | \(e\left(\frac{2909}{4004}\right)\) | \(e\left(\frac{305}{572}\right)\) | \(e\left(\frac{997}{2002}\right)\) | \(e\left(\frac{69}{143}\right)\) | \(e\left(\frac{1753}{4004}\right)\) | \(e\left(\frac{787}{4004}\right)\) |
| \(\chi_{10015}(187,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{359}{572}\right)\) | \(e\left(\frac{2987}{4004}\right)\) | \(e\left(\frac{73}{286}\right)\) | \(e\left(\frac{34}{91}\right)\) | \(e\left(\frac{625}{4004}\right)\) | \(e\left(\frac{505}{572}\right)\) | \(e\left(\frac{985}{2002}\right)\) | \(e\left(\frac{133}{143}\right)\) | \(e\left(\frac{5}{4004}\right)\) | \(e\left(\frac{2147}{4004}\right)\) |
| \(\chi_{10015}(188,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{249}{572}\right)\) | \(e\left(\frac{633}{4004}\right)\) | \(e\left(\frac{249}{286}\right)\) | \(e\left(\frac{54}{91}\right)\) | \(e\left(\frac{2363}{4004}\right)\) | \(e\left(\frac{175}{572}\right)\) | \(e\left(\frac{633}{2002}\right)\) | \(e\left(\frac{56}{143}\right)\) | \(e\left(\frac{115}{4004}\right)\) | \(e\left(\frac{1333}{4004}\right)\) |
| \(\chi_{10015}(192,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{383}{572}\right)\) | \(e\left(\frac{3875}{4004}\right)\) | \(e\left(\frac{97}{286}\right)\) | \(e\left(\frac{58}{91}\right)\) | \(e\left(\frac{1473}{4004}\right)\) | \(e\left(\frac{5}{572}\right)\) | \(e\left(\frac{1873}{2002}\right)\) | \(e\left(\frac{116}{143}\right)\) | \(e\left(\frac{1229}{4004}\right)\) | \(e\left(\frac{1607}{4004}\right)\) |
| \(\chi_{10015}(197,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{563}{572}\right)\) | \(e\left(\frac{3099}{4004}\right)\) | \(e\left(\frac{277}{286}\right)\) | \(e\left(\frac{69}{91}\right)\) | \(e\left(\frac{3257}{4004}\right)\) | \(e\left(\frac{545}{572}\right)\) | \(e\left(\frac{1097}{2002}\right)\) | \(e\left(\frac{60}{143}\right)\) | \(e\left(\frac{2973}{4004}\right)\) | \(e\left(\frac{131}{4004}\right)\) |
| \(\chi_{10015}(198,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{289}{572}\right)\) | \(e\left(\frac{1541}{4004}\right)\) | \(e\left(\frac{3}{286}\right)\) | \(e\left(\frac{81}{91}\right)\) | \(e\left(\frac{1679}{4004}\right)\) | \(e\left(\frac{295}{572}\right)\) | \(e\left(\frac{1541}{2002}\right)\) | \(e\left(\frac{123}{143}\right)\) | \(e\left(\frac{1583}{4004}\right)\) | \(e\left(\frac{3865}{4004}\right)\) |
| \(\chi_{10015}(203,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{229}{572}\right)\) | \(e\left(\frac{1609}{4004}\right)\) | \(e\left(\frac{229}{286}\right)\) | \(e\left(\frac{73}{91}\right)\) | \(e\left(\frac{1275}{4004}\right)\) | \(e\left(\frac{115}{572}\right)\) | \(e\left(\frac{1609}{2002}\right)\) | \(e\left(\frac{94}{143}\right)\) | \(e\left(\frac{811}{4004}\right)\) | \(e\left(\frac{1497}{4004}\right)\) |
| \(\chi_{10015}(212,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{327}{572}\right)\) | \(e\left(\frac{1231}{4004}\right)\) | \(e\left(\frac{41}{286}\right)\) | \(e\left(\frac{80}{91}\right)\) | \(e\left(\frac{1401}{4004}\right)\) | \(e\left(\frac{409}{572}\right)\) | \(e\left(\frac{1231}{2002}\right)\) | \(e\left(\frac{108}{143}\right)\) | \(e\left(\frac{1805}{4004}\right)\) | \(e\left(\frac{2295}{4004}\right)\) |
| \(\chi_{10015}(217,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{479}{572}\right)\) | \(e\left(\frac{2279}{4004}\right)\) | \(e\left(\frac{193}{286}\right)\) | \(e\left(\frac{37}{91}\right)\) | \(e\left(\frac{2005}{4004}\right)\) | \(e\left(\frac{293}{572}\right)\) | \(e\left(\frac{277}{2002}\right)\) | \(e\left(\frac{48}{143}\right)\) | \(e\left(\frac{977}{4004}\right)\) | \(e\left(\frac{2307}{4004}\right)\) |
| \(\chi_{10015}(218,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{449}{572}\right)\) | \(e\left(\frac{25}{4004}\right)\) | \(e\left(\frac{163}{286}\right)\) | \(e\left(\frac{72}{91}\right)\) | \(e\left(\frac{87}{4004}\right)\) | \(e\left(\frac{203}{572}\right)\) | \(e\left(\frac{25}{2002}\right)\) | \(e\left(\frac{105}{143}\right)\) | \(e\left(\frac{2307}{4004}\right)\) | \(e\left(\frac{837}{4004}\right)\) |
| \(\chi_{10015}(222,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{159}{572}\right)\) | \(e\left(\frac{1307}{4004}\right)\) | \(e\left(\frac{159}{286}\right)\) | \(e\left(\frac{55}{91}\right)\) | \(e\left(\frac{1185}{4004}\right)\) | \(e\left(\frac{477}{572}\right)\) | \(e\left(\frac{1307}{2002}\right)\) | \(e\left(\frac{84}{143}\right)\) | \(e\left(\frac{3533}{4004}\right)\) | \(e\left(\frac{355}{4004}\right)\) |
| \(\chi_{10015}(223,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{225}{572}\right)\) | \(e\left(\frac{3177}{4004}\right)\) | \(e\left(\frac{225}{286}\right)\) | \(e\left(\frac{17}{91}\right)\) | \(e\left(\frac{2087}{4004}\right)\) | \(e\left(\frac{103}{572}\right)\) | \(e\left(\frac{1175}{2002}\right)\) | \(e\left(\frac{73}{143}\right)\) | \(e\left(\frac{2323}{4004}\right)\) | \(e\left(\frac{1301}{4004}\right)\) |