from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4027, base_ring=CyclotomicField(4026))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,4027))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4027\) | |
| Conductor: | \(4027\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(4026\) | 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: | $\Q(\zeta_{2013})$ |
| Fixed field: | Number field defined by a degree 4026 polynomial (not computed) |
First 31 of 1200 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4027}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{1342}\right)\) | \(e\left(\frac{1}{4026}\right)\) | \(e\left(\frac{125}{671}\right)\) | \(e\left(\frac{47}{4026}\right)\) | \(e\left(\frac{188}{2013}\right)\) | \(e\left(\frac{607}{1342}\right)\) | \(e\left(\frac{375}{1342}\right)\) | \(e\left(\frac{1}{2013}\right)\) | \(e\left(\frac{211}{2013}\right)\) | \(e\left(\frac{3875}{4026}\right)\) |
| \(\chi_{4027}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{507}{1342}\right)\) | \(e\left(\frac{47}{4026}\right)\) | \(e\left(\frac{507}{671}\right)\) | \(e\left(\frac{2209}{4026}\right)\) | \(e\left(\frac{784}{2013}\right)\) | \(e\left(\frac{347}{1342}\right)\) | \(e\left(\frac{179}{1342}\right)\) | \(e\left(\frac{47}{2013}\right)\) | \(e\left(\frac{1865}{2013}\right)\) | \(e\left(\frac{955}{4026}\right)\) |
| \(\chi_{4027}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1255}{1342}\right)\) | \(e\left(\frac{3875}{4026}\right)\) | \(e\left(\frac{584}{671}\right)\) | \(e\left(\frac{955}{4026}\right)\) | \(e\left(\frac{1807}{2013}\right)\) | \(e\left(\frac{941}{1342}\right)\) | \(e\left(\frac{1081}{1342}\right)\) | \(e\left(\frac{1862}{2013}\right)\) | \(e\left(\frac{347}{2013}\right)\) | \(e\left(\frac{2671}{4026}\right)\) |
| \(\chi_{4027}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1277}{1342}\right)\) | \(e\left(\frac{751}{4026}\right)\) | \(e\left(\frac{606}{671}\right)\) | \(e\left(\frac{3089}{4026}\right)\) | \(e\left(\frac{278}{2013}\right)\) | \(e\left(\frac{919}{1342}\right)\) | \(e\left(\frac{1147}{1342}\right)\) | \(e\left(\frac{751}{2013}\right)\) | \(e\left(\frac{1447}{2013}\right)\) | \(e\left(\frac{3353}{4026}\right)\) |
| \(\chi_{4027}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{155}{1342}\right)\) | \(e\left(\frac{377}{4026}\right)\) | \(e\left(\frac{155}{671}\right)\) | \(e\left(\frac{1615}{4026}\right)\) | \(e\left(\frac{421}{2013}\right)\) | \(e\left(\frac{699}{1342}\right)\) | \(e\left(\frac{465}{1342}\right)\) | \(e\left(\frac{377}{2013}\right)\) | \(e\left(\frac{1040}{2013}\right)\) | \(e\left(\frac{3463}{4026}\right)\) |
| \(\chi_{4027}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{317}{1342}\right)\) | \(e\left(\frac{797}{4026}\right)\) | \(e\left(\frac{317}{671}\right)\) | \(e\left(\frac{1225}{4026}\right)\) | \(e\left(\frac{874}{2013}\right)\) | \(e\left(\frac{659}{1342}\right)\) | \(e\left(\frac{951}{1342}\right)\) | \(e\left(\frac{797}{2013}\right)\) | \(e\left(\frac{1088}{2013}\right)\) | \(e\left(\frac{433}{4026}\right)\) |
| \(\chi_{4027}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{857}{1342}\right)\) | \(e\left(\frac{3539}{4026}\right)\) | \(e\left(\frac{186}{671}\right)\) | \(e\left(\frac{1267}{4026}\right)\) | \(e\left(\frac{1042}{2013}\right)\) | \(e\left(\frac{973}{1342}\right)\) | \(e\left(\frac{1229}{1342}\right)\) | \(e\left(\frac{1526}{2013}\right)\) | \(e\left(\frac{1919}{2013}\right)\) | \(e\left(\frac{1069}{4026}\right)\) |
| \(\chi_{4027}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{535}{1342}\right)\) | \(e\left(\frac{1561}{4026}\right)\) | \(e\left(\frac{535}{671}\right)\) | \(e\left(\frac{899}{4026}\right)\) | \(e\left(\frac{1583}{2013}\right)\) | \(e\left(\frac{75}{1342}\right)\) | \(e\left(\frac{263}{1342}\right)\) | \(e\left(\frac{1561}{2013}\right)\) | \(e\left(\frac{1252}{2013}\right)\) | \(e\left(\frac{1823}{4026}\right)\) |
| \(\chi_{4027}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{1342}\right)\) | \(e\left(\frac{773}{4026}\right)\) | \(e\left(\frac{1}{671}\right)\) | \(e\left(\frac{97}{4026}\right)\) | \(e\left(\frac{388}{2013}\right)\) | \(e\left(\frac{853}{1342}\right)\) | \(e\left(\frac{3}{1342}\right)\) | \(e\left(\frac{773}{2013}\right)\) | \(e\left(\frac{50}{2013}\right)\) | \(e\left(\frac{31}{4026}\right)\) |
| \(\chi_{4027}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{555}{1342}\right)\) | \(e\left(\frac{917}{4026}\right)\) | \(e\left(\frac{555}{671}\right)\) | \(e\left(\frac{2839}{4026}\right)\) | \(e\left(\frac{1291}{2013}\right)\) | \(e\left(\frac{1031}{1342}\right)\) | \(e\left(\frac{323}{1342}\right)\) | \(e\left(\frac{917}{2013}\right)\) | \(e\left(\frac{239}{2013}\right)\) | \(e\left(\frac{2443}{4026}\right)\) |
| \(\chi_{4027}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{323}{1342}\right)\) | \(e\left(\frac{67}{4026}\right)\) | \(e\left(\frac{323}{671}\right)\) | \(e\left(\frac{3149}{4026}\right)\) | \(e\left(\frac{518}{2013}\right)\) | \(e\left(\frac{409}{1342}\right)\) | \(e\left(\frac{969}{1342}\right)\) | \(e\left(\frac{67}{2013}\right)\) | \(e\left(\frac{46}{2013}\right)\) | \(e\left(\frac{1961}{4026}\right)\) |
| \(\chi_{4027}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{857}{1342}\right)\) | \(e\left(\frac{2197}{4026}\right)\) | \(e\left(\frac{186}{671}\right)\) | \(e\left(\frac{2609}{4026}\right)\) | \(e\left(\frac{371}{2013}\right)\) | \(e\left(\frac{973}{1342}\right)\) | \(e\left(\frac{1229}{1342}\right)\) | \(e\left(\frac{184}{2013}\right)\) | \(e\left(\frac{577}{2013}\right)\) | \(e\left(\frac{2411}{4026}\right)\) |
| \(\chi_{4027}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1065}{1342}\right)\) | \(e\left(\frac{599}{4026}\right)\) | \(e\left(\frac{394}{671}\right)\) | \(e\left(\frac{3997}{4026}\right)\) | \(e\left(\frac{1897}{2013}\right)\) | \(e\left(\frac{1253}{1342}\right)\) | \(e\left(\frac{511}{1342}\right)\) | \(e\left(\frac{599}{2013}\right)\) | \(e\left(\frac{1583}{2013}\right)\) | \(e\left(\frac{2149}{4026}\right)\) |
| \(\chi_{4027}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{757}{1342}\right)\) | \(e\left(\frac{49}{4026}\right)\) | \(e\left(\frac{86}{671}\right)\) | \(e\left(\frac{2303}{4026}\right)\) | \(e\left(\frac{1160}{2013}\right)\) | \(e\left(\frac{219}{1342}\right)\) | \(e\left(\frac{929}{1342}\right)\) | \(e\left(\frac{49}{2013}\right)\) | \(e\left(\frac{274}{2013}\right)\) | \(e\left(\frac{653}{4026}\right)\) |
| \(\chi_{4027}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1087}{1342}\right)\) | \(e\left(\frac{1501}{4026}\right)\) | \(e\left(\frac{416}{671}\right)\) | \(e\left(\frac{2105}{4026}\right)\) | \(e\left(\frac{368}{2013}\right)\) | \(e\left(\frac{1231}{1342}\right)\) | \(e\left(\frac{577}{1342}\right)\) | \(e\left(\frac{1501}{2013}\right)\) | \(e\left(\frac{670}{2013}\right)\) | \(e\left(\frac{2831}{4026}\right)\) |
| \(\chi_{4027}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{919}{1342}\right)\) | \(e\left(\frac{469}{4026}\right)\) | \(e\left(\frac{248}{671}\right)\) | \(e\left(\frac{1913}{4026}\right)\) | \(e\left(\frac{1613}{2013}\right)\) | \(e\left(\frac{179}{1342}\right)\) | \(e\left(\frac{73}{1342}\right)\) | \(e\left(\frac{469}{2013}\right)\) | \(e\left(\frac{322}{2013}\right)\) | \(e\left(\frac{1649}{4026}\right)\) |
| \(\chi_{4027}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{755}{1342}\right)\) | \(e\left(\frac{1187}{4026}\right)\) | \(e\left(\frac{84}{671}\right)\) | \(e\left(\frac{3451}{4026}\right)\) | \(e\left(\frac{1726}{2013}\right)\) | \(e\left(\frac{1197}{1342}\right)\) | \(e\left(\frac{923}{1342}\right)\) | \(e\left(\frac{1187}{2013}\right)\) | \(e\left(\frac{845}{2013}\right)\) | \(e\left(\frac{1933}{4026}\right)\) |
| \(\chi_{4027}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{141}{1342}\right)\) | \(e\left(\frac{1633}{4026}\right)\) | \(e\left(\frac{141}{671}\right)\) | \(e\left(\frac{257}{4026}\right)\) | \(e\left(\frac{1028}{2013}\right)\) | \(e\left(\frac{835}{1342}\right)\) | \(e\left(\frac{423}{1342}\right)\) | \(e\left(\frac{1633}{2013}\right)\) | \(e\left(\frac{340}{2013}\right)\) | \(e\left(\frac{3029}{4026}\right)\) |
| \(\chi_{4027}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{863}{1342}\right)\) | \(e\left(\frac{125}{4026}\right)\) | \(e\left(\frac{192}{671}\right)\) | \(e\left(\frac{1849}{4026}\right)\) | \(e\left(\frac{1357}{2013}\right)\) | \(e\left(\frac{723}{1342}\right)\) | \(e\left(\frac{1247}{1342}\right)\) | \(e\left(\frac{125}{2013}\right)\) | \(e\left(\frac{206}{2013}\right)\) | \(e\left(\frac{1255}{4026}\right)\) |
| \(\chi_{4027}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1077}{1342}\right)\) | \(e\left(\frac{1823}{4026}\right)\) | \(e\left(\frac{406}{671}\right)\) | \(e\left(\frac{1135}{4026}\right)\) | \(e\left(\frac{514}{2013}\right)\) | \(e\left(\frac{753}{1342}\right)\) | \(e\left(\frac{547}{1342}\right)\) | \(e\left(\frac{1823}{2013}\right)\) | \(e\left(\frac{170}{2013}\right)\) | \(e\left(\frac{2521}{4026}\right)\) |
| \(\chi_{4027}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{705}{1342}\right)\) | \(e\left(\frac{113}{4026}\right)\) | \(e\left(\frac{34}{671}\right)\) | \(e\left(\frac{1285}{4026}\right)\) | \(e\left(\frac{1114}{2013}\right)\) | \(e\left(\frac{149}{1342}\right)\) | \(e\left(\frac{773}{1342}\right)\) | \(e\left(\frac{113}{2013}\right)\) | \(e\left(\frac{1700}{2013}\right)\) | \(e\left(\frac{3067}{4026}\right)\) |
| \(\chi_{4027}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1239}{1342}\right)\) | \(e\left(\frac{2243}{4026}\right)\) | \(e\left(\frac{568}{671}\right)\) | \(e\left(\frac{745}{4026}\right)\) | \(e\left(\frac{967}{2013}\right)\) | \(e\left(\frac{713}{1342}\right)\) | \(e\left(\frac{1033}{1342}\right)\) | \(e\left(\frac{230}{2013}\right)\) | \(e\left(\frac{218}{2013}\right)\) | \(e\left(\frac{3517}{4026}\right)\) |
| \(\chi_{4027}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1307}{1342}\right)\) | \(e\left(\frac{1127}{4026}\right)\) | \(e\left(\frac{636}{671}\right)\) | \(e\left(\frac{631}{4026}\right)\) | \(e\left(\frac{511}{2013}\right)\) | \(e\left(\frac{1011}{1342}\right)\) | \(e\left(\frac{1237}{1342}\right)\) | \(e\left(\frac{1127}{2013}\right)\) | \(e\left(\frac{263}{2013}\right)\) | \(e\left(\frac{2941}{4026}\right)\) |
| \(\chi_{4027}(75,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1139}{1342}\right)\) | \(e\left(\frac{95}{4026}\right)\) | \(e\left(\frac{468}{671}\right)\) | \(e\left(\frac{439}{4026}\right)\) | \(e\left(\frac{1756}{2013}\right)\) | \(e\left(\frac{1301}{1342}\right)\) | \(e\left(\frac{733}{1342}\right)\) | \(e\left(\frac{95}{2013}\right)\) | \(e\left(\frac{1928}{2013}\right)\) | \(e\left(\frac{1759}{4026}\right)\) |
| \(\chi_{4027}(80,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{1342}\right)\) | \(e\left(\frac{1547}{4026}\right)\) | \(e\left(\frac{127}{671}\right)\) | \(e\left(\frac{241}{4026}\right)\) | \(e\left(\frac{964}{2013}\right)\) | \(e\left(\frac{971}{1342}\right)\) | \(e\left(\frac{381}{1342}\right)\) | \(e\left(\frac{1547}{2013}\right)\) | \(e\left(\frac{311}{2013}\right)\) | \(e\left(\frac{3937}{4026}\right)\) |
| \(\chi_{4027}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{609}{1342}\right)\) | \(e\left(\frac{2399}{4026}\right)\) | \(e\left(\frac{609}{671}\right)\) | \(e\left(\frac{25}{4026}\right)\) | \(e\left(\frac{100}{2013}\right)\) | \(e\left(\frac{123}{1342}\right)\) | \(e\left(\frac{485}{1342}\right)\) | \(e\left(\frac{386}{2013}\right)\) | \(e\left(\frac{926}{2013}\right)\) | \(e\left(\frac{91}{4026}\right)\) |
| \(\chi_{4027}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{361}{1342}\right)\) | \(e\left(\frac{1259}{4026}\right)\) | \(e\left(\frac{361}{671}\right)\) | \(e\left(\frac{2809}{4026}\right)\) | \(e\left(\frac{1171}{2013}\right)\) | \(e\left(\frac{615}{1342}\right)\) | \(e\left(\frac{1083}{1342}\right)\) | \(e\left(\frac{1259}{2013}\right)\) | \(e\left(\frac{1946}{2013}\right)\) | \(e\left(\frac{3139}{4026}\right)\) |
| \(\chi_{4027}(92,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1229}{1342}\right)\) | \(e\left(\frac{1223}{4026}\right)\) | \(e\left(\frac{558}{671}\right)\) | \(e\left(\frac{1117}{4026}\right)\) | \(e\left(\frac{442}{2013}\right)\) | \(e\left(\frac{235}{1342}\right)\) | \(e\left(\frac{1003}{1342}\right)\) | \(e\left(\frac{1223}{2013}\right)\) | \(e\left(\frac{389}{2013}\right)\) | \(e\left(\frac{523}{4026}\right)\) |
| \(\chi_{4027}(95,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1157}{1342}\right)\) | \(e\left(\frac{589}{4026}\right)\) | \(e\left(\frac{486}{671}\right)\) | \(e\left(\frac{3527}{4026}\right)\) | \(e\left(\frac{17}{2013}\right)\) | \(e\left(\frac{551}{1342}\right)\) | \(e\left(\frac{787}{1342}\right)\) | \(e\left(\frac{589}{2013}\right)\) | \(e\left(\frac{1486}{2013}\right)\) | \(e\left(\frac{3659}{4026}\right)\) |
| \(\chi_{4027}(99,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{1342}\right)\) | \(e\left(\frac{3877}{4026}\right)\) | \(e\left(\frac{163}{671}\right)\) | \(e\left(\frac{1049}{4026}\right)\) | \(e\left(\frac{170}{2013}\right)\) | \(e\left(\frac{813}{1342}\right)\) | \(e\left(\frac{489}{1342}\right)\) | \(e\left(\frac{1864}{2013}\right)\) | \(e\left(\frac{769}{2013}\right)\) | \(e\left(\frac{2369}{4026}\right)\) |
| \(\chi_{4027}(106,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{787}{1342}\right)\) | \(e\left(\frac{3109}{4026}\right)\) | \(e\left(\frac{116}{671}\right)\) | \(e\left(\frac{1187}{4026}\right)\) | \(e\left(\frac{722}{2013}\right)\) | \(e\left(\frac{311}{1342}\right)\) | \(e\left(\frac{1019}{1342}\right)\) | \(e\left(\frac{1096}{2013}\right)\) | \(e\left(\frac{1774}{2013}\right)\) | \(e\left(\frac{1583}{4026}\right)\) |