from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4013, base_ring=CyclotomicField(2006))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,4013))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4013\) | |
| Conductor: | \(4013\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2006\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{1003})$ |
| Fixed field: | Number field defined by a degree 2006 polynomial (not computed) |
First 31 of 928 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4013}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{2006}\right)\) | \(e\left(\frac{1745}{2006}\right)\) | \(e\left(\frac{1}{1003}\right)\) | \(e\left(\frac{235}{2006}\right)\) | \(e\left(\frac{873}{1003}\right)\) | \(e\left(\frac{303}{1003}\right)\) | \(e\left(\frac{3}{2006}\right)\) | \(e\left(\frac{742}{1003}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{83}{1003}\right)\) |
| \(\chi_{4013}(6,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{873}{2006}\right)\) | \(e\left(\frac{831}{2006}\right)\) | \(e\left(\frac{873}{1003}\right)\) | \(e\left(\frac{543}{2006}\right)\) | \(e\left(\frac{852}{1003}\right)\) | \(e\left(\frac{730}{1003}\right)\) | \(e\left(\frac{613}{2006}\right)\) | \(e\left(\frac{831}{1003}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{243}{1003}\right)\) |
| \(\chi_{4013}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1745}{2006}\right)\) | \(e\left(\frac{1923}{2006}\right)\) | \(e\left(\frac{742}{1003}\right)\) | \(e\left(\frac{851}{2006}\right)\) | \(e\left(\frac{831}{1003}\right)\) | \(e\left(\frac{154}{1003}\right)\) | \(e\left(\frac{1223}{2006}\right)\) | \(e\left(\frac{920}{1003}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{403}{1003}\right)\) |
| \(\chi_{4013}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{355}{2006}\right)\) | \(e\left(\frac{1627}{2006}\right)\) | \(e\left(\frac{355}{1003}\right)\) | \(e\left(\frac{1179}{2006}\right)\) | \(e\left(\frac{991}{1003}\right)\) | \(e\left(\frac{244}{1003}\right)\) | \(e\left(\frac{1065}{2006}\right)\) | \(e\left(\frac{624}{1003}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{378}{1003}\right)\) |
| \(\chi_{4013}(15,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1993}{2006}\right)\) | \(e\left(\frac{1387}{2006}\right)\) | \(e\left(\frac{990}{1003}\right)\) | \(e\left(\frac{957}{2006}\right)\) | \(e\left(\frac{687}{1003}\right)\) | \(e\left(\frac{73}{1003}\right)\) | \(e\left(\frac{1967}{2006}\right)\) | \(e\left(\frac{384}{1003}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{927}{1003}\right)\) |
| \(\chi_{4013}(17,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1329}{2006}\right)\) | \(e\left(\frac{169}{2006}\right)\) | \(e\left(\frac{326}{1003}\right)\) | \(e\left(\frac{1385}{2006}\right)\) | \(e\left(\frac{749}{1003}\right)\) | \(e\left(\frac{484}{1003}\right)\) | \(e\left(\frac{1981}{2006}\right)\) | \(e\left(\frac{169}{1003}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{980}{1003}\right)\) |
| \(\chi_{4013}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{235}{2006}\right)\) | \(e\left(\frac{851}{2006}\right)\) | \(e\left(\frac{235}{1003}\right)\) | \(e\left(\frac{1063}{2006}\right)\) | \(e\left(\frac{543}{1003}\right)\) | \(e\left(\frac{995}{1003}\right)\) | \(e\left(\frac{705}{2006}\right)\) | \(e\left(\frac{851}{1003}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{448}{1003}\right)\) |
| \(\chi_{4013}(28,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1307}{2006}\right)\) | \(e\left(\frac{1899}{2006}\right)\) | \(e\left(\frac{304}{1003}\right)\) | \(e\left(\frac{227}{2006}\right)\) | \(e\left(\frac{600}{1003}\right)\) | \(e\left(\frac{839}{1003}\right)\) | \(e\left(\frac{1915}{2006}\right)\) | \(e\left(\frac{896}{1003}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{157}{1003}\right)\) |
| \(\chi_{4013}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1859}{2006}\right)\) | \(e\left(\frac{253}{2006}\right)\) | \(e\left(\frac{856}{1003}\right)\) | \(e\left(\frac{1563}{2006}\right)\) | \(e\left(\frac{53}{1003}\right)\) | \(e\left(\frac{594}{1003}\right)\) | \(e\left(\frac{1565}{2006}\right)\) | \(e\left(\frac{253}{1003}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{838}{1003}\right)\) |
| \(\chi_{4013}(42,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{2006}\right)\) | \(e\left(\frac{985}{2006}\right)\) | \(e\left(\frac{173}{1003}\right)\) | \(e\left(\frac{535}{2006}\right)\) | \(e\left(\frac{579}{1003}\right)\) | \(e\left(\frac{263}{1003}\right)\) | \(e\left(\frac{519}{2006}\right)\) | \(e\left(\frac{985}{1003}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{317}{1003}\right)\) |
| \(\chi_{4013}(44,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1087}{2006}\right)\) | \(e\left(\frac{1145}{2006}\right)\) | \(e\left(\frac{84}{1003}\right)\) | \(e\left(\frac{683}{2006}\right)\) | \(e\left(\frac{113}{1003}\right)\) | \(e\left(\frac{377}{1003}\right)\) | \(e\left(\frac{1255}{2006}\right)\) | \(e\left(\frac{142}{1003}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{954}{1003}\right)\) |
| \(\chi_{4013}(46,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1339}{2006}\right)\) | \(e\left(\frac{1571}{2006}\right)\) | \(e\left(\frac{336}{1003}\right)\) | \(e\left(\frac{1729}{2006}\right)\) | \(e\left(\frac{452}{1003}\right)\) | \(e\left(\frac{505}{1003}\right)\) | \(e\left(\frac{5}{2006}\right)\) | \(e\left(\frac{568}{1003}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{807}{1003}\right)\) |
| \(\chi_{4013}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{299}{2006}\right)\) | \(e\left(\frac{195}{2006}\right)\) | \(e\left(\frac{299}{1003}\right)\) | \(e\left(\frac{55}{2006}\right)\) | \(e\left(\frac{247}{1003}\right)\) | \(e\left(\frac{327}{1003}\right)\) | \(e\left(\frac{897}{2006}\right)\) | \(e\left(\frac{195}{1003}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{745}{1003}\right)\) |
| \(\chi_{4013}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1679}{2006}\right)\) | \(e\left(\frac{1095}{2006}\right)\) | \(e\left(\frac{676}{1003}\right)\) | \(e\left(\frac{1389}{2006}\right)\) | \(e\left(\frac{384}{1003}\right)\) | \(e\left(\frac{216}{1003}\right)\) | \(e\left(\frac{1025}{2006}\right)\) | \(e\left(\frac{92}{1003}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{943}{1003}\right)\) |
| \(\chi_{4013}(63,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1045}{2006}\right)\) | \(e\left(\frac{71}{2006}\right)\) | \(e\left(\frac{42}{1003}\right)\) | \(e\left(\frac{843}{2006}\right)\) | \(e\left(\frac{558}{1003}\right)\) | \(e\left(\frac{690}{1003}\right)\) | \(e\left(\frac{1129}{2006}\right)\) | \(e\left(\frac{71}{1003}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{477}{1003}\right)\) |
| \(\chi_{4013}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{2006}\right)\) | \(e\left(\frac{1223}{2006}\right)\) | \(e\left(\frac{3}{1003}\right)\) | \(e\left(\frac{705}{2006}\right)\) | \(e\left(\frac{613}{1003}\right)\) | \(e\left(\frac{909}{1003}\right)\) | \(e\left(\frac{9}{2006}\right)\) | \(e\left(\frac{220}{1003}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{249}{1003}\right)\) |
| \(\chi_{4013}(66,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1959}{2006}\right)\) | \(e\left(\frac{231}{2006}\right)\) | \(e\left(\frac{956}{1003}\right)\) | \(e\left(\frac{991}{2006}\right)\) | \(e\left(\frac{92}{1003}\right)\) | \(e\left(\frac{804}{1003}\right)\) | \(e\left(\frac{1865}{2006}\right)\) | \(e\left(\frac{231}{1003}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{111}{1003}\right)\) |
| \(\chi_{4013}(69,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{205}{2006}\right)\) | \(e\left(\frac{657}{2006}\right)\) | \(e\left(\frac{205}{1003}\right)\) | \(e\left(\frac{31}{2006}\right)\) | \(e\left(\frac{431}{1003}\right)\) | \(e\left(\frac{932}{1003}\right)\) | \(e\left(\frac{615}{2006}\right)\) | \(e\left(\frac{657}{1003}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{967}{1003}\right)\) |
| \(\chi_{4013}(70,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{421}{2006}\right)\) | \(e\left(\frac{449}{2006}\right)\) | \(e\left(\frac{421}{1003}\right)\) | \(e\left(\frac{641}{2006}\right)\) | \(e\left(\frac{435}{1003}\right)\) | \(e\left(\frac{182}{1003}\right)\) | \(e\left(\frac{1263}{2006}\right)\) | \(e\left(\frac{449}{1003}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{841}{1003}\right)\) |
| \(\chi_{4013}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1415}{2006}\right)\) | \(e\left(\frac{1795}{2006}\right)\) | \(e\left(\frac{412}{1003}\right)\) | \(e\left(\frac{1535}{2006}\right)\) | \(e\left(\frac{602}{1003}\right)\) | \(e\left(\frac{464}{1003}\right)\) | \(e\left(\frac{233}{2006}\right)\) | \(e\left(\frac{792}{1003}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{94}{1003}\right)\) |
| \(\chi_{4013}(76,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1049}{2006}\right)\) | \(e\left(\frac{1033}{2006}\right)\) | \(e\left(\frac{46}{1003}\right)\) | \(e\left(\frac{1783}{2006}\right)\) | \(e\left(\frac{38}{1003}\right)\) | \(e\left(\frac{899}{1003}\right)\) | \(e\left(\frac{1141}{2006}\right)\) | \(e\left(\frac{30}{1003}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{809}{1003}\right)\) |
| \(\chi_{4013}(89,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{2006}\right)\) | \(e\left(\frac{269}{2006}\right)\) | \(e\left(\frac{145}{1003}\right)\) | \(e\left(\frac{1979}{2006}\right)\) | \(e\left(\frac{207}{1003}\right)\) | \(e\left(\frac{806}{1003}\right)\) | \(e\left(\frac{435}{2006}\right)\) | \(e\left(\frac{269}{1003}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{1002}{1003}\right)\) |
| \(\chi_{4013}(91,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1661}{2006}\right)\) | \(e\left(\frac{1781}{2006}\right)\) | \(e\left(\frac{658}{1003}\right)\) | \(e\left(\frac{1171}{2006}\right)\) | \(e\left(\frac{718}{1003}\right)\) | \(e\left(\frac{780}{1003}\right)\) | \(e\left(\frac{971}{2006}\right)\) | \(e\left(\frac{778}{1003}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{452}{1003}\right)\) |
| \(\chi_{4013}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{875}{2006}\right)\) | \(e\left(\frac{309}{2006}\right)\) | \(e\left(\frac{875}{1003}\right)\) | \(e\left(\frac{1013}{2006}\right)\) | \(e\left(\frac{592}{1003}\right)\) | \(e\left(\frac{333}{1003}\right)\) | \(e\left(\frac{619}{2006}\right)\) | \(e\left(\frac{309}{1003}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{409}{1003}\right)\) |
| \(\chi_{4013}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{2006}\right)\) | \(e\left(\frac{411}{2006}\right)\) | \(e\left(\frac{229}{1003}\right)\) | \(e\left(\frac{1659}{2006}\right)\) | \(e\left(\frac{320}{1003}\right)\) | \(e\left(\frac{180}{1003}\right)\) | \(e\left(\frac{687}{2006}\right)\) | \(e\left(\frac{411}{1003}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{953}{1003}\right)\) |
| \(\chi_{4013}(99,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{825}{2006}\right)\) | \(e\left(\frac{1323}{2006}\right)\) | \(e\left(\frac{825}{1003}\right)\) | \(e\left(\frac{1299}{2006}\right)\) | \(e\left(\frac{71}{1003}\right)\) | \(e\left(\frac{228}{1003}\right)\) | \(e\left(\frac{469}{2006}\right)\) | \(e\left(\frac{320}{1003}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{271}{1003}\right)\) |
| \(\chi_{4013}(105,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1293}{2006}\right)\) | \(e\left(\frac{1541}{2006}\right)\) | \(e\left(\frac{290}{1003}\right)\) | \(e\left(\frac{949}{2006}\right)\) | \(e\left(\frac{414}{1003}\right)\) | \(e\left(\frac{609}{1003}\right)\) | \(e\left(\frac{1873}{2006}\right)\) | \(e\left(\frac{538}{1003}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{1001}{1003}\right)\) |
| \(\chi_{4013}(110,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{201}{2006}\right)\) | \(e\left(\frac{1701}{2006}\right)\) | \(e\left(\frac{201}{1003}\right)\) | \(e\left(\frac{1097}{2006}\right)\) | \(e\left(\frac{951}{1003}\right)\) | \(e\left(\frac{723}{1003}\right)\) | \(e\left(\frac{603}{2006}\right)\) | \(e\left(\frac{698}{1003}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{635}{1003}\right)\) |
| \(\chi_{4013}(115,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{453}{2006}\right)\) | \(e\left(\frac{121}{2006}\right)\) | \(e\left(\frac{453}{1003}\right)\) | \(e\left(\frac{137}{2006}\right)\) | \(e\left(\frac{287}{1003}\right)\) | \(e\left(\frac{851}{1003}\right)\) | \(e\left(\frac{1359}{2006}\right)\) | \(e\left(\frac{121}{1003}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{488}{1003}\right)\) |
| \(\chi_{4013}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{2006}\right)\) | \(e\left(\frac{1783}{2006}\right)\) | \(e\left(\frac{193}{1003}\right)\) | \(e\left(\frac{1223}{2006}\right)\) | \(e\left(\frac{988}{1003}\right)\) | \(e\left(\frac{305}{1003}\right)\) | \(e\left(\frac{579}{2006}\right)\) | \(e\left(\frac{780}{1003}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{974}{1003}\right)\) |
| \(\chi_{4013}(143,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1441}{2006}\right)\) | \(e\left(\frac{1027}{2006}\right)\) | \(e\left(\frac{438}{1003}\right)\) | \(e\left(\frac{1627}{2006}\right)\) | \(e\left(\frac{231}{1003}\right)\) | \(e\left(\frac{318}{1003}\right)\) | \(e\left(\frac{311}{2006}\right)\) | \(e\left(\frac{24}{1003}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{246}{1003}\right)\) |