from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10015, base_ring=CyclotomicField(4004))
M = H._module
chi = DirichletCharacter(H, M([1001,246]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,10015))
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: | even | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{10015}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{225}{572}\right)\) | \(e\left(\frac{31}{4004}\right)\) | \(e\left(\frac{225}{286}\right)\) | \(e\left(\frac{73}{182}\right)\) | \(e\left(\frac{3231}{4004}\right)\) | \(e\left(\frac{103}{572}\right)\) | \(e\left(\frac{31}{2002}\right)\) | \(e\left(\frac{3}{286}\right)\) | \(e\left(\frac{3181}{4004}\right)\) | \(e\left(\frac{2159}{4004}\right)\) |
| \(\chi_{10015}(18,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{447}{572}\right)\) | \(e\left(\frac{3669}{4004}\right)\) | \(e\left(\frac{161}{286}\right)\) | \(e\left(\frac{127}{182}\right)\) | \(e\left(\frac{1637}{4004}\right)\) | \(e\left(\frac{197}{572}\right)\) | \(e\left(\frac{1667}{2002}\right)\) | \(e\left(\frac{189}{286}\right)\) | \(e\left(\frac{1919}{4004}\right)\) | \(e\left(\frac{1597}{4004}\right)\) |
| \(\chi_{10015}(28,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{387}{572}\right)\) | \(e\left(\frac{2593}{4004}\right)\) | \(e\left(\frac{101}{286}\right)\) | \(e\left(\frac{59}{182}\right)\) | \(e\left(\frac{2377}{4004}\right)\) | \(e\left(\frac{17}{572}\right)\) | \(e\left(\frac{591}{2002}\right)\) | \(e\left(\frac{131}{286}\right)\) | \(e\left(\frac{3}{4004}\right)\) | \(e\left(\frac{2089}{4004}\right)\) |
| \(\chi_{10015}(33,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{311}{572}\right)\) | \(e\left(\frac{2069}{4004}\right)\) | \(e\left(\frac{25}{286}\right)\) | \(e\left(\frac{11}{182}\right)\) | \(e\left(\frac{73}{4004}\right)\) | \(e\left(\frac{361}{572}\right)\) | \(e\left(\frac{67}{2002}\right)\) | \(e\left(\frac{191}{286}\right)\) | \(e\left(\frac{2419}{4004}\right)\) | \(e\left(\frac{81}{4004}\right)\) |
| \(\chi_{10015}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{181}{572}\right)\) | \(e\left(\frac{1835}{4004}\right)\) | \(e\left(\frac{181}{286}\right)\) | \(e\left(\frac{141}{182}\right)\) | \(e\left(\frac{3583}{4004}\right)\) | \(e\left(\frac{543}{572}\right)\) | \(e\left(\frac{1835}{2002}\right)\) | \(e\left(\frac{113}{286}\right)\) | \(e\left(\frac{365}{4004}\right)\) | \(e\left(\frac{575}{4004}\right)\) |
| \(\chi_{10015}(38,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{535}{572}\right)\) | \(e\left(\frac{1205}{4004}\right)\) | \(e\left(\frac{249}{286}\right)\) | \(e\left(\frac{43}{182}\right)\) | \(e\left(\frac{3793}{4004}\right)\) | \(e\left(\frac{461}{572}\right)\) | \(e\left(\frac{1205}{2002}\right)\) | \(e\left(\frac{255}{286}\right)\) | \(e\left(\frac{687}{4004}\right)\) | \(e\left(\frac{1905}{4004}\right)\) |
| \(\chi_{10015}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{503}{572}\right)\) | \(e\left(\frac{593}{4004}\right)\) | \(e\left(\frac{217}{286}\right)\) | \(e\left(\frac{5}{182}\right)\) | \(e\left(\frac{3425}{4004}\right)\) | \(e\left(\frac{365}{572}\right)\) | \(e\left(\frac{593}{2002}\right)\) | \(e\left(\frac{205}{286}\right)\) | \(e\left(\frac{3631}{4004}\right)\) | \(e\left(\frac{3197}{4004}\right)\) |
| \(\chi_{10015}(63,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{572}\right)\) | \(e\left(\frac{417}{4004}\right)\) | \(e\left(\frac{19}{286}\right)\) | \(e\left(\frac{25}{182}\right)\) | \(e\left(\frac{3293}{4004}\right)\) | \(e\left(\frac{57}{572}\right)\) | \(e\left(\frac{417}{2002}\right)\) | \(e\left(\frac{271}{286}\right)\) | \(e\left(\frac{683}{4004}\right)\) | \(e\left(\frac{1789}{4004}\right)\) |
| \(\chi_{10015}(68,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{572}\right)\) | \(e\left(\frac{3673}{4004}\right)\) | \(e\left(\frac{107}{286}\right)\) | \(e\left(\frac{19}{182}\right)\) | \(e\left(\frac{3733}{4004}\right)\) | \(e\left(\frac{321}{572}\right)\) | \(e\left(\frac{1671}{2002}\right)\) | \(e\left(\frac{51}{286}\right)\) | \(e\left(\frac{1167}{4004}\right)\) | \(e\left(\frac{3813}{4004}\right)\) |
| \(\chi_{10015}(72,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{572}\right)\) | \(e\left(\frac{2227}{4004}\right)\) | \(e\left(\frac{37}{286}\right)\) | \(e\left(\frac{113}{182}\right)\) | \(e\left(\frac{783}{4004}\right)\) | \(e\left(\frac{111}{572}\right)\) | \(e\left(\frac{225}{2002}\right)\) | \(e\left(\frac{31}{286}\right)\) | \(e\left(\frac{2745}{4004}\right)\) | \(e\left(\frac{1527}{4004}\right)\) |
| \(\chi_{10015}(78,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{259}{572}\right)\) | \(e\left(\frac{1289}{4004}\right)\) | \(e\left(\frac{259}{286}\right)\) | \(e\left(\frac{141}{182}\right)\) | \(e\left(\frac{3765}{4004}\right)\) | \(e\left(\frac{205}{572}\right)\) | \(e\left(\frac{1289}{2002}\right)\) | \(e\left(\frac{217}{286}\right)\) | \(e\left(\frac{911}{4004}\right)\) | \(e\left(\frac{393}{4004}\right)\) |
| \(\chi_{10015}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{572}\right)\) | \(e\left(\frac{2845}{4004}\right)\) | \(e\left(\frac{131}{286}\right)\) | \(e\left(\frac{171}{182}\right)\) | \(e\left(\frac{2293}{4004}\right)\) | \(e\left(\frac{393}{572}\right)\) | \(e\left(\frac{843}{2002}\right)\) | \(e\left(\frac{17}{286}\right)\) | \(e\left(\frac{675}{4004}\right)\) | \(e\left(\frac{1557}{4004}\right)\) |
| \(\chi_{10015}(93,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{572}\right)\) | \(e\left(\frac{2441}{4004}\right)\) | \(e\left(\frac{151}{286}\right)\) | \(e\left(\frac{159}{182}\right)\) | \(e\left(\frac{2809}{4004}\right)\) | \(e\left(\frac{453}{572}\right)\) | \(e\left(\frac{439}{2002}\right)\) | \(e\left(\frac{227}{286}\right)\) | \(e\left(\frac{551}{4004}\right)\) | \(e\left(\frac{1965}{4004}\right)\) |
| \(\chi_{10015}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{333}{572}\right)\) | \(e\left(\frac{1167}{4004}\right)\) | \(e\left(\frac{47}{286}\right)\) | \(e\left(\frac{159}{182}\right)\) | \(e\left(\frac{1899}{4004}\right)\) | \(e\left(\frac{427}{572}\right)\) | \(e\left(\frac{1167}{2002}\right)\) | \(e\left(\frac{279}{286}\right)\) | \(e\left(\frac{1825}{4004}\right)\) | \(e\left(\frac{2875}{4004}\right)\) |
| \(\chi_{10015}(98,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{531}{572}\right)\) | \(e\left(\frac{3345}{4004}\right)\) | \(e\left(\frac{245}{286}\right)\) | \(e\left(\frac{139}{182}\right)\) | \(e\left(\frac{29}{4004}\right)\) | \(e\left(\frac{449}{572}\right)\) | \(e\left(\frac{1343}{2002}\right)\) | \(e\left(\frac{213}{286}\right)\) | \(e\left(\frac{2771}{4004}\right)\) | \(e\left(\frac{2281}{4004}\right)\) |
| \(\chi_{10015}(103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{379}{572}\right)\) | \(e\left(\frac{2869}{4004}\right)\) | \(e\left(\frac{93}{286}\right)\) | \(e\left(\frac{69}{182}\right)\) | \(e\left(\frac{2857}{4004}\right)\) | \(e\left(\frac{565}{572}\right)\) | \(e\left(\frac{867}{2002}\right)\) | \(e\left(\frac{47}{286}\right)\) | \(e\left(\frac{167}{4004}\right)\) | \(e\left(\frac{2841}{4004}\right)\) |
| \(\chi_{10015}(112,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{549}{572}\right)\) | \(e\left(\frac{1151}{4004}\right)\) | \(e\left(\frac{263}{286}\right)\) | \(e\left(\frac{45}{182}\right)\) | \(e\left(\frac{1523}{4004}\right)\) | \(e\left(\frac{503}{572}\right)\) | \(e\left(\frac{1151}{2002}\right)\) | \(e\left(\frac{259}{286}\right)\) | \(e\left(\frac{829}{4004}\right)\) | \(e\left(\frac{2019}{4004}\right)\) |
| \(\chi_{10015}(118,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{572}\right)\) | \(e\left(\frac{2085}{4004}\right)\) | \(e\left(\frac{95}{286}\right)\) | \(e\left(\frac{125}{182}\right)\) | \(e\left(\frac{449}{4004}\right)\) | \(e\left(\frac{285}{572}\right)\) | \(e\left(\frac{83}{2002}\right)\) | \(e\left(\frac{211}{286}\right)\) | \(e\left(\frac{3415}{4004}\right)\) | \(e\left(\frac{937}{4004}\right)\) |
| \(\chi_{10015}(123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{283}{572}\right)\) | \(e\left(\frac{1605}{4004}\right)\) | \(e\left(\frac{283}{286}\right)\) | \(e\left(\frac{163}{182}\right)\) | \(e\left(\frac{1181}{4004}\right)\) | \(e\left(\frac{277}{572}\right)\) | \(e\left(\frac{1605}{2002}\right)\) | \(e\left(\frac{183}{286}\right)\) | \(e\left(\frac{1563}{4004}\right)\) | \(e\left(\frac{3285}{4004}\right)\) |
| \(\chi_{10015}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{572}\right)\) | \(e\left(\frac{1459}{4004}\right)\) | \(e\left(\frac{109}{286}\right)\) | \(e\left(\frac{101}{182}\right)\) | \(e\left(\frac{2755}{4004}\right)\) | \(e\left(\frac{327}{572}\right)\) | \(e\left(\frac{1459}{2002}\right)\) | \(e\left(\frac{215}{286}\right)\) | \(e\left(\frac{2985}{4004}\right)\) | \(e\left(\frac{479}{4004}\right)\) |
| \(\chi_{10015}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{572}\right)\) | \(e\left(\frac{1957}{4004}\right)\) | \(e\left(\frac{107}{286}\right)\) | \(e\left(\frac{123}{182}\right)\) | \(e\left(\frac{1445}{4004}\right)\) | \(e\left(\frac{321}{572}\right)\) | \(e\left(\frac{1957}{2002}\right)\) | \(e\left(\frac{51}{286}\right)\) | \(e\left(\frac{3455}{4004}\right)\) | \(e\left(\frac{2097}{4004}\right)\) |
| \(\chi_{10015}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{572}\right)\) | \(e\left(\frac{491}{4004}\right)\) | \(e\left(\frac{21}{286}\right)\) | \(e\left(\frac{29}{182}\right)\) | \(e\left(\frac{27}{4004}\right)\) | \(e\left(\frac{63}{572}\right)\) | \(e\left(\frac{491}{2002}\right)\) | \(e\left(\frac{149}{286}\right)\) | \(e\left(\frac{785}{4004}\right)\) | \(e\left(\frac{743}{4004}\right)\) |
| \(\chi_{10015}(143,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{123}{572}\right)\) | \(e\left(\frac{3693}{4004}\right)\) | \(e\left(\frac{123}{286}\right)\) | \(e\left(\frac{25}{182}\right)\) | \(e\left(\frac{2201}{4004}\right)\) | \(e\left(\frac{369}{572}\right)\) | \(e\left(\frac{1691}{2002}\right)\) | \(e\left(\frac{219}{286}\right)\) | \(e\left(\frac{1411}{4004}\right)\) | \(e\left(\frac{2881}{4004}\right)\) |
| \(\chi_{10015}(148,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{343}{572}\right)\) | \(e\left(\frac{393}{4004}\right)\) | \(e\left(\frac{57}{286}\right)\) | \(e\left(\frac{127}{182}\right)\) | \(e\left(\frac{2729}{4004}\right)\) | \(e\left(\frac{457}{572}\right)\) | \(e\left(\frac{393}{2002}\right)\) | \(e\left(\frac{241}{286}\right)\) | \(e\left(\frac{1191}{4004}\right)\) | \(e\left(\frac{505}{4004}\right)\) |
| \(\chi_{10015}(152,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{572}\right)\) | \(e\left(\frac{3767}{4004}\right)\) | \(e\left(\frac{125}{286}\right)\) | \(e\left(\frac{29}{182}\right)\) | \(e\left(\frac{2939}{4004}\right)\) | \(e\left(\frac{375}{572}\right)\) | \(e\left(\frac{1765}{2002}\right)\) | \(e\left(\frac{97}{286}\right)\) | \(e\left(\frac{1513}{4004}\right)\) | \(e\left(\frac{1835}{4004}\right)\) |
| \(\chi_{10015}(153,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{311}{572}\right)\) | \(e\left(\frac{1497}{4004}\right)\) | \(e\left(\frac{25}{286}\right)\) | \(e\left(\frac{167}{182}\right)\) | \(e\left(\frac{645}{4004}\right)\) | \(e\left(\frac{361}{572}\right)\) | \(e\left(\frac{1497}{2002}\right)\) | \(e\left(\frac{191}{286}\right)\) | \(e\left(\frac{1847}{4004}\right)\) | \(e\left(\frac{3513}{4004}\right)\) |
| \(\chi_{10015}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{397}{572}\right)\) | \(e\left(\frac{2391}{4004}\right)\) | \(e\left(\frac{111}{286}\right)\) | \(e\left(\frac{53}{182}\right)\) | \(e\left(\frac{2635}{4004}\right)\) | \(e\left(\frac{47}{572}\right)\) | \(e\left(\frac{389}{2002}\right)\) | \(e\left(\frac{93}{286}\right)\) | \(e\left(\frac{3945}{4004}\right)\) | \(e\left(\frac{291}{4004}\right)\) |
| \(\chi_{10015}(158,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{419}{572}\right)\) | \(e\left(\frac{1489}{4004}\right)\) | \(e\left(\frac{133}{286}\right)\) | \(e\left(\frac{19}{182}\right)\) | \(e\left(\frac{457}{4004}\right)\) | \(e\left(\frac{113}{572}\right)\) | \(e\left(\frac{1489}{2002}\right)\) | \(e\left(\frac{181}{286}\right)\) | \(e\left(\frac{3351}{4004}\right)\) | \(e\left(\frac{3085}{4004}\right)\) |
| \(\chi_{10015}(162,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{241}{572}\right)\) | \(e\left(\frac{51}{4004}\right)\) | \(e\left(\frac{241}{286}\right)\) | \(e\left(\frac{79}{182}\right)\) | \(e\left(\frac{1699}{4004}\right)\) | \(e\left(\frac{151}{572}\right)\) | \(e\left(\frac{51}{2002}\right)\) | \(e\left(\frac{171}{286}\right)\) | \(e\left(\frac{3425}{4004}\right)\) | \(e\left(\frac{1227}{4004}\right)\) |
| \(\chi_{10015}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{171}{572}\right)\) | \(e\left(\frac{321}{4004}\right)\) | \(e\left(\frac{171}{286}\right)\) | \(e\left(\frac{69}{182}\right)\) | \(e\left(\frac{1037}{4004}\right)\) | \(e\left(\frac{513}{572}\right)\) | \(e\left(\frac{321}{2002}\right)\) | \(e\left(\frac{151}{286}\right)\) | \(e\left(\frac{2715}{4004}\right)\) | \(e\left(\frac{657}{4004}\right)\) |
| \(\chi_{10015}(172,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{572}\right)\) | \(e\left(\frac{3155}{4004}\right)\) | \(e\left(\frac{93}{286}\right)\) | \(e\left(\frac{173}{182}\right)\) | \(e\left(\frac{2571}{4004}\right)\) | \(e\left(\frac{279}{572}\right)\) | \(e\left(\frac{1153}{2002}\right)\) | \(e\left(\frac{47}{286}\right)\) | \(e\left(\frac{453}{4004}\right)\) | \(e\left(\frac{3127}{4004}\right)\) |