from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4889, base_ring=CyclotomicField(4888))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,4889))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(4889\) | |
Conductor: | \(4889\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(4888\) | 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: | $\Q(\zeta_{4888})$ |
Fixed field: | Number field defined by a degree 4888 polynomial (not computed) |
First 31 of 2208 characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{4889}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{617}{2444}\right)\) | \(e\left(\frac{1}{4888}\right)\) | \(e\left(\frac{617}{1222}\right)\) | \(e\left(\frac{579}{1222}\right)\) | \(e\left(\frac{95}{376}\right)\) | \(e\left(\frac{889}{4888}\right)\) | \(e\left(\frac{1851}{2444}\right)\) | \(e\left(\frac{1}{2444}\right)\) | \(e\left(\frac{1775}{2444}\right)\) | \(e\left(\frac{1995}{2444}\right)\) |
\(\chi_{4889}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1057}{2444}\right)\) | \(e\left(\frac{889}{4888}\right)\) | \(e\left(\frac{1057}{1222}\right)\) | \(e\left(\frac{269}{1222}\right)\) | \(e\left(\frac{231}{376}\right)\) | \(e\left(\frac{3353}{4888}\right)\) | \(e\left(\frac{727}{2444}\right)\) | \(e\left(\frac{889}{2444}\right)\) | \(e\left(\frac{1595}{2444}\right)\) | \(e\left(\frac{1655}{2444}\right)\) |
\(\chi_{4889}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{761}{2444}\right)\) | \(e\left(\frac{2469}{4888}\right)\) | \(e\left(\frac{761}{1222}\right)\) | \(e\left(\frac{1033}{1222}\right)\) | \(e\left(\frac{307}{376}\right)\) | \(e\left(\frac{229}{4888}\right)\) | \(e\left(\frac{2283}{2444}\right)\) | \(e\left(\frac{25}{2444}\right)\) | \(e\left(\frac{383}{2444}\right)\) | \(e\left(\frac{995}{2444}\right)\) |
\(\chi_{4889}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2351}{2444}\right)\) | \(e\left(\frac{2123}{4888}\right)\) | \(e\left(\frac{1129}{1222}\right)\) | \(e\left(\frac{1107}{1222}\right)\) | \(e\left(\frac{149}{376}\right)\) | \(e\left(\frac{579}{4888}\right)\) | \(e\left(\frac{2165}{2444}\right)\) | \(e\left(\frac{2123}{2444}\right)\) | \(e\left(\frac{2121}{2444}\right)\) | \(e\left(\frac{2377}{2444}\right)\) |
\(\chi_{4889}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2293}{2444}\right)\) | \(e\left(\frac{2317}{4888}\right)\) | \(e\left(\frac{1071}{1222}\right)\) | \(e\left(\frac{1009}{1222}\right)\) | \(e\left(\frac{155}{376}\right)\) | \(e\left(\frac{1965}{4888}\right)\) | \(e\left(\frac{1991}{2444}\right)\) | \(e\left(\frac{2317}{2444}\right)\) | \(e\left(\frac{1867}{2444}\right)\) | \(e\left(\frac{811}{2444}\right)\) |
\(\chi_{4889}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{685}{2444}\right)\) | \(e\left(\frac{4493}{4888}\right)\) | \(e\left(\frac{685}{1222}\right)\) | \(e\left(\frac{1031}{1222}\right)\) | \(e\left(\frac{75}{376}\right)\) | \(e\left(\frac{781}{4888}\right)\) | \(e\left(\frac{2055}{2444}\right)\) | \(e\left(\frac{2049}{2444}\right)\) | \(e\left(\frac{303}{2444}\right)\) | \(e\left(\frac{1387}{2444}\right)\) |
\(\chi_{4889}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2055}{2444}\right)\) | \(e\left(\frac{3703}{4888}\right)\) | \(e\left(\frac{833}{1222}\right)\) | \(e\left(\frac{649}{1222}\right)\) | \(e\left(\frac{225}{376}\right)\) | \(e\left(\frac{2343}{4888}\right)\) | \(e\left(\frac{1277}{2444}\right)\) | \(e\left(\frac{1259}{2444}\right)\) | \(e\left(\frac{909}{2444}\right)\) | \(e\left(\frac{1717}{2444}\right)\) |
\(\chi_{4889}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1851}{2444}\right)\) | \(e\left(\frac{3}{4888}\right)\) | \(e\left(\frac{629}{1222}\right)\) | \(e\left(\frac{515}{1222}\right)\) | \(e\left(\frac{285}{376}\right)\) | \(e\left(\frac{2667}{4888}\right)\) | \(e\left(\frac{665}{2444}\right)\) | \(e\left(\frac{3}{2444}\right)\) | \(e\left(\frac{437}{2444}\right)\) | \(e\left(\frac{1097}{2444}\right)\) |
\(\chi_{4889}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1201}{2444}\right)\) | \(e\left(\frac{3357}{4888}\right)\) | \(e\left(\frac{1201}{1222}\right)\) | \(e\left(\frac{723}{1222}\right)\) | \(e\left(\frac{67}{376}\right)\) | \(e\left(\frac{2693}{4888}\right)\) | \(e\left(\frac{1159}{2444}\right)\) | \(e\left(\frac{913}{2444}\right)\) | \(e\left(\frac{203}{2444}\right)\) | \(e\left(\frac{655}{2444}\right)\) |
\(\chi_{4889}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{2444}\right)\) | \(e\left(\frac{2345}{4888}\right)\) | \(e\left(\frac{17}{1222}\right)\) | \(e\left(\frac{113}{1222}\right)\) | \(e\left(\frac{183}{376}\right)\) | \(e\left(\frac{2417}{4888}\right)\) | \(e\left(\frac{51}{2444}\right)\) | \(e\left(\frac{2345}{2444}\right)\) | \(e\left(\frac{243}{2444}\right)\) | \(e\left(\frac{459}{2444}\right)\) |
\(\chi_{4889}(30,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1143}{2444}\right)\) | \(e\left(\frac{3551}{4888}\right)\) | \(e\left(\frac{1143}{1222}\right)\) | \(e\left(\frac{625}{1222}\right)\) | \(e\left(\frac{73}{376}\right)\) | \(e\left(\frac{4079}{4888}\right)\) | \(e\left(\frac{985}{2444}\right)\) | \(e\left(\frac{1107}{2444}\right)\) | \(e\left(\frac{2393}{2444}\right)\) | \(e\left(\frac{1533}{2444}\right)\) |
\(\chi_{4889}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{2444}\right)\) | \(e\left(\frac{1743}{4888}\right)\) | \(e\left(\frac{71}{1222}\right)\) | \(e\left(\frac{1047}{1222}\right)\) | \(e\left(\frac{145}{376}\right)\) | \(e\left(\frac{31}{4888}\right)\) | \(e\left(\frac{213}{2444}\right)\) | \(e\left(\frac{1743}{2444}\right)\) | \(e\left(\frac{2165}{2444}\right)\) | \(e\left(\frac{1917}{2444}\right)\) |
\(\chi_{4889}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1979}{2444}\right)\) | \(e\left(\frac{839}{4888}\right)\) | \(e\left(\frac{757}{1222}\right)\) | \(e\left(\frac{647}{1222}\right)\) | \(e\left(\frac{369}{376}\right)\) | \(e\left(\frac{2895}{4888}\right)\) | \(e\left(\frac{1049}{2444}\right)\) | \(e\left(\frac{839}{2444}\right)\) | \(e\left(\frac{829}{2444}\right)\) | \(e\left(\frac{2109}{2444}\right)\) |
\(\chi_{4889}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{289}{2444}\right)\) | \(e\left(\frac{3205}{4888}\right)\) | \(e\left(\frac{289}{1222}\right)\) | \(e\left(\frac{699}{1222}\right)\) | \(e\left(\frac{291}{376}\right)\) | \(e\left(\frac{4429}{4888}\right)\) | \(e\left(\frac{867}{2444}\right)\) | \(e\left(\frac{761}{2444}\right)\) | \(e\left(\frac{1687}{2444}\right)\) | \(e\left(\frac{471}{2444}\right)\) |
\(\chi_{4889}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{2444}\right)\) | \(e\left(\frac{3189}{4888}\right)\) | \(e\left(\frac{193}{1222}\right)\) | \(e\left(\frac{1211}{1222}\right)\) | \(e\left(\frac{275}{376}\right)\) | \(e\left(\frac{4869}{4888}\right)\) | \(e\left(\frac{579}{2444}\right)\) | \(e\left(\frac{745}{2444}\right)\) | \(e\left(\frac{171}{2444}\right)\) | \(e\left(\frac{323}{2444}\right)\) |
\(\chi_{4889}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2211}{2444}\right)\) | \(e\left(\frac{2507}{4888}\right)\) | \(e\left(\frac{989}{1222}\right)\) | \(e\left(\frac{1039}{1222}\right)\) | \(e\left(\frac{157}{376}\right)\) | \(e\left(\frac{4683}{4888}\right)\) | \(e\left(\frac{1745}{2444}\right)\) | \(e\left(\frac{63}{2444}\right)\) | \(e\left(\frac{1845}{2444}\right)\) | \(e\left(\frac{1041}{2444}\right)\) |
\(\chi_{4889}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1795}{2444}\right)\) | \(e\left(\frac{4067}{4888}\right)\) | \(e\left(\frac{573}{1222}\right)\) | \(e\left(\frac{1221}{1222}\right)\) | \(e\left(\frac{213}{376}\right)\) | \(e\left(\frac{3331}{4888}\right)\) | \(e\left(\frac{497}{2444}\right)\) | \(e\left(\frac{1623}{2444}\right)\) | \(e\left(\frac{1793}{2444}\right)\) | \(e\left(\frac{2029}{2444}\right)\) |
\(\chi_{4889}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{905}{2444}\right)\) | \(e\left(\frac{49}{4888}\right)\) | \(e\left(\frac{905}{1222}\right)\) | \(e\left(\frac{265}{1222}\right)\) | \(e\left(\frac{143}{376}\right)\) | \(e\left(\frac{4457}{4888}\right)\) | \(e\left(\frac{271}{2444}\right)\) | \(e\left(\frac{49}{2444}\right)\) | \(e\left(\frac{1435}{2444}\right)\) | \(e\left(\frac{2439}{2444}\right)\) |
\(\chi_{4889}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{701}{2444}\right)\) | \(e\left(\frac{1237}{4888}\right)\) | \(e\left(\frac{701}{1222}\right)\) | \(e\left(\frac{131}{1222}\right)\) | \(e\left(\frac{203}{376}\right)\) | \(e\left(\frac{4781}{4888}\right)\) | \(e\left(\frac{2103}{2444}\right)\) | \(e\left(\frac{1237}{2444}\right)\) | \(e\left(\frac{963}{2444}\right)\) | \(e\left(\frac{1819}{2444}\right)\) |
\(\chi_{4889}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{2444}\right)\) | \(e\left(\frac{4591}{4888}\right)\) | \(e\left(\frac{51}{1222}\right)\) | \(e\left(\frac{339}{1222}\right)\) | \(e\left(\frac{361}{376}\right)\) | \(e\left(\frac{4807}{4888}\right)\) | \(e\left(\frac{153}{2444}\right)\) | \(e\left(\frac{2147}{2444}\right)\) | \(e\left(\frac{729}{2444}\right)\) | \(e\left(\frac{1377}{2444}\right)\) |
\(\chi_{4889}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1311}{2444}\right)\) | \(e\left(\frac{3579}{4888}\right)\) | \(e\left(\frac{89}{1222}\right)\) | \(e\left(\frac{951}{1222}\right)\) | \(e\left(\frac{101}{376}\right)\) | \(e\left(\frac{4531}{4888}\right)\) | \(e\left(\frac{1489}{2444}\right)\) | \(e\left(\frac{1135}{2444}\right)\) | \(e\left(\frac{769}{2444}\right)\) | \(e\left(\frac{1181}{2444}\right)\) |
\(\chi_{4889}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2437}{2444}\right)\) | \(e\left(\frac{4785}{4888}\right)\) | \(e\left(\frac{1215}{1222}\right)\) | \(e\left(\frac{241}{1222}\right)\) | \(e\left(\frac{367}{376}\right)\) | \(e\left(\frac{1305}{4888}\right)\) | \(e\left(\frac{2423}{2444}\right)\) | \(e\left(\frac{2341}{2444}\right)\) | \(e\left(\frac{475}{2444}\right)\) | \(e\left(\frac{2255}{2444}\right)\) |
\(\chi_{4889}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2291}{2444}\right)\) | \(e\left(\frac{891}{4888}\right)\) | \(e\left(\frac{1069}{1222}\right)\) | \(e\left(\frac{205}{1222}\right)\) | \(e\left(\frac{45}{376}\right)\) | \(e\left(\frac{243}{4888}\right)\) | \(e\left(\frac{1985}{2444}\right)\) | \(e\left(\frac{891}{2444}\right)\) | \(e\left(\frac{257}{2444}\right)\) | \(e\left(\frac{757}{2444}\right)\) |
\(\chi_{4889}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{189}{2444}\right)\) | \(e\left(\frac{337}{4888}\right)\) | \(e\left(\frac{189}{1222}\right)\) | \(e\left(\frac{825}{1222}\right)\) | \(e\left(\frac{55}{376}\right)\) | \(e\left(\frac{1425}{4888}\right)\) | \(e\left(\frac{567}{2444}\right)\) | \(e\left(\frac{337}{2444}\right)\) | \(e\left(\frac{1839}{2444}\right)\) | \(e\left(\frac{215}{2444}\right)\) |
\(\chi_{4889}(68,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{829}{2444}\right)\) | \(e\left(\frac{2073}{4888}\right)\) | \(e\left(\frac{829}{1222}\right)\) | \(e\left(\frac{263}{1222}\right)\) | \(e\left(\frac{287}{376}\right)\) | \(e\left(\frac{121}{4888}\right)\) | \(e\left(\frac{43}{2444}\right)\) | \(e\left(\frac{2073}{2444}\right)\) | \(e\left(\frac{1355}{2444}\right)\) | \(e\left(\frac{387}{2444}\right)\) |
\(\chi_{4889}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2085}{2444}\right)\) | \(e\left(\frac{3097}{4888}\right)\) | \(e\left(\frac{863}{1222}\right)\) | \(e\left(\frac{489}{1222}\right)\) | \(e\left(\frac{183}{376}\right)\) | \(e\left(\frac{1289}{4888}\right)\) | \(e\left(\frac{1367}{2444}\right)\) | \(e\left(\frac{653}{2444}\right)\) | \(e\left(\frac{619}{2444}\right)\) | \(e\left(\frac{83}{2444}\right)\) |
\(\chi_{4889}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1583}{2444}\right)\) | \(e\left(\frac{4439}{4888}\right)\) | \(e\left(\frac{361}{1222}\right)\) | \(e\left(\frac{315}{1222}\right)\) | \(e\left(\frac{209}{376}\right)\) | \(e\left(\frac{1655}{4888}\right)\) | \(e\left(\frac{2305}{2444}\right)\) | \(e\left(\frac{1995}{2444}\right)\) | \(e\left(\frac{2213}{2444}\right)\) | \(e\left(\frac{1193}{2444}\right)\) |
\(\chi_{4889}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{885}{2444}\right)\) | \(e\left(\frac{453}{4888}\right)\) | \(e\left(\frac{885}{1222}\right)\) | \(e\left(\frac{779}{1222}\right)\) | \(e\left(\frac{171}{376}\right)\) | \(e\left(\frac{1901}{4888}\right)\) | \(e\left(\frac{211}{2444}\right)\) | \(e\left(\frac{453}{2444}\right)\) | \(e\left(\frac{2443}{2444}\right)\) | \(e\left(\frac{1899}{2444}\right)\) |
\(\chi_{4889}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1487}{2444}\right)\) | \(e\left(\frac{4423}{4888}\right)\) | \(e\left(\frac{265}{1222}\right)\) | \(e\left(\frac{827}{1222}\right)\) | \(e\left(\frac{193}{376}\right)\) | \(e\left(\frac{2095}{4888}\right)\) | \(e\left(\frac{2017}{2444}\right)\) | \(e\left(\frac{1979}{2444}\right)\) | \(e\left(\frac{697}{2444}\right)\) | \(e\left(\frac{1045}{2444}\right)\) |
\(\chi_{4889}(75,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1525}{2444}\right)\) | \(e\left(\frac{4633}{4888}\right)\) | \(e\left(\frac{303}{1222}\right)\) | \(e\left(\frac{217}{1222}\right)\) | \(e\left(\frac{215}{376}\right)\) | \(e\left(\frac{3041}{4888}\right)\) | \(e\left(\frac{2131}{2444}\right)\) | \(e\left(\frac{2189}{2444}\right)\) | \(e\left(\frac{1959}{2444}\right)\) | \(e\left(\frac{2071}{2444}\right)\) |
\(\chi_{4889}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1779}{2444}\right)\) | \(e\left(\frac{4879}{4888}\right)\) | \(e\left(\frac{557}{1222}\right)\) | \(e\left(\frac{899}{1222}\right)\) | \(e\left(\frac{273}{376}\right)\) | \(e\left(\frac{1775}{4888}\right)\) | \(e\left(\frac{449}{2444}\right)\) | \(e\left(\frac{2435}{2444}\right)\) | \(e\left(\frac{1133}{2444}\right)\) | \(e\left(\frac{1597}{2444}\right)\) |