from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1003, base_ring=CyclotomicField(116))
M = H._module
chi = DirichletCharacter(H, M([29,90]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,1003))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1003\) | |
| Conductor: | \(1003\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(116\) | 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: | $\Q(\zeta_{116})$ |
| Fixed field: | Number field defined by a degree 116 polynomial (not computed) |
First 31 of 56 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1003}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{5}{116}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{105}{116}\right)\) | \(e\left(\frac{37}{116}\right)\) | \(e\left(\frac{83}{116}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{21}{116}\right)\) | \(e\left(\frac{17}{116}\right)\) |
| \(\chi_{1003}(30,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{45}{116}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{17}{116}\right)\) | \(e\left(\frac{101}{116}\right)\) | \(e\left(\frac{51}{116}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{73}{116}\right)\) | \(e\left(\frac{37}{116}\right)\) |
| \(\chi_{1003}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{43}{116}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{91}{116}\right)\) | \(e\left(\frac{63}{116}\right)\) | \(e\left(\frac{41}{116}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{43}{58}\right)\) | \(e\left(\frac{111}{116}\right)\) | \(e\left(\frac{7}{116}\right)\) |
| \(\chi_{1003}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{9}{116}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{73}{116}\right)\) | \(e\left(\frac{113}{116}\right)\) | \(e\left(\frac{103}{116}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{9}{58}\right)\) | \(e\left(\frac{61}{116}\right)\) | \(e\left(\frac{77}{116}\right)\) |
| \(\chi_{1003}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{55}{116}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{111}{116}\right)\) | \(e\left(\frac{59}{116}\right)\) | \(e\left(\frac{101}{116}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{55}{58}\right)\) | \(e\left(\frac{115}{116}\right)\) | \(e\left(\frac{71}{116}\right)\) |
| \(\chi_{1003}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{63}{116}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{47}{116}\right)\) | \(e\left(\frac{95}{116}\right)\) | \(e\left(\frac{25}{116}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{79}{116}\right)\) | \(e\left(\frac{75}{116}\right)\) |
| \(\chi_{1003}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{103}{116}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{75}{116}\right)\) | \(e\left(\frac{43}{116}\right)\) | \(e\left(\frac{109}{116}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{15}{116}\right)\) | \(e\left(\frac{95}{116}\right)\) |
| \(\chi_{1003}(98,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{17}{116}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{9}{116}\right)\) | \(e\left(\frac{33}{116}\right)\) | \(e\left(\frac{27}{116}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{17}{58}\right)\) | \(e\left(\frac{25}{116}\right)\) | \(e\left(\frac{81}{116}\right)\) |
| \(\chi_{1003}(106,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{67}{116}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{15}{116}\right)\) | \(e\left(\frac{55}{116}\right)\) | \(e\left(\frac{45}{116}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{9}{58}\right)\) | \(e\left(\frac{3}{116}\right)\) | \(e\left(\frac{19}{116}\right)\) |
| \(\chi_{1003}(115,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{41}{116}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{49}{116}\right)\) | \(e\left(\frac{25}{116}\right)\) | \(e\left(\frac{31}{116}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{33}{116}\right)\) | \(e\left(\frac{93}{116}\right)\) |
| \(\chi_{1003}(132,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{73}{116}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{25}{116}\right)\) | \(e\left(\frac{53}{116}\right)\) | \(e\left(\frac{75}{116}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{15}{58}\right)\) | \(e\left(\frac{5}{116}\right)\) | \(e\left(\frac{109}{116}\right)\) |
| \(\chi_{1003}(149,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{57}{116}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{37}{116}\right)\) | \(e\left(\frac{97}{116}\right)\) | \(e\left(\frac{111}{116}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{77}{116}\right)\) | \(e\left(\frac{101}{116}\right)\) |
| \(\chi_{1003}(157,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{75}{116}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{67}{116}\right)\) | \(e\left(\frac{91}{116}\right)\) | \(e\left(\frac{85}{116}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{17}{58}\right)\) | \(e\left(\frac{83}{116}\right)\) | \(e\left(\frac{23}{116}\right)\) |
| \(\chi_{1003}(174,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{99}{116}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{107}{116}\right)\) | \(e\left(\frac{83}{116}\right)\) | \(e\left(\frac{89}{116}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{91}{116}\right)\) | \(e\left(\frac{35}{116}\right)\) |
| \(\chi_{1003}(183,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{25}{116}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{61}{116}\right)\) | \(e\left(\frac{69}{116}\right)\) | \(e\left(\frac{67}{116}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{105}{116}\right)\) | \(e\left(\frac{85}{116}\right)\) |
| \(\chi_{1003}(191,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{15}{116}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{83}{116}\right)\) | \(e\left(\frac{111}{116}\right)\) | \(e\left(\frac{17}{116}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{15}{58}\right)\) | \(e\left(\frac{63}{116}\right)\) | \(e\left(\frac{51}{116}\right)\) |
| \(\chi_{1003}(200,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{21}{116}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{93}{116}\right)\) | \(e\left(\frac{109}{116}\right)\) | \(e\left(\frac{47}{116}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{65}{116}\right)\) | \(e\left(\frac{25}{116}\right)\) |
| \(\chi_{1003}(208,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{115}{116}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{95}{116}\right)\) | \(e\left(\frac{39}{116}\right)\) | \(e\left(\frac{53}{116}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{19}{116}\right)\) | \(e\left(\frac{43}{116}\right)\) |
| \(\chi_{1003}(217,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{1}{116}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{21}{116}\right)\) | \(e\left(\frac{77}{116}\right)\) | \(e\left(\frac{63}{116}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{97}{116}\right)\) | \(e\left(\frac{73}{116}\right)\) |
| \(\chi_{1003}(242,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{83}{116}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{3}{116}\right)\) | \(e\left(\frac{11}{116}\right)\) | \(e\left(\frac{9}{116}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{47}{116}\right)\) | \(e\left(\frac{27}{116}\right)\) |
| \(\chi_{1003}(259,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{79}{116}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{35}{116}\right)\) | \(e\left(\frac{51}{116}\right)\) | \(e\left(\frac{105}{116}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{7}{116}\right)\) | \(e\left(\frac{83}{116}\right)\) |
| \(\chi_{1003}(268,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{65}{116}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{89}{116}\right)\) | \(e\left(\frac{17}{116}\right)\) | \(e\left(\frac{35}{116}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{41}{116}\right)\) | \(e\left(\frac{105}{116}\right)\) |
| \(\chi_{1003}(276,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{59}{116}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{79}{116}\right)\) | \(e\left(\frac{19}{116}\right)\) | \(e\left(\frac{5}{116}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{39}{116}\right)\) | \(e\left(\frac{15}{116}\right)\) |
| \(\chi_{1003}(319,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{109}{116}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{85}{116}\right)\) | \(e\left(\frac{41}{116}\right)\) | \(e\left(\frac{23}{116}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{17}{116}\right)\) | \(e\left(\frac{69}{116}\right)\) |
| \(\chi_{1003}(327,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{7}{116}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{31}{116}\right)\) | \(e\left(\frac{75}{116}\right)\) | \(e\left(\frac{93}{116}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{99}{116}\right)\) | \(e\left(\frac{47}{116}\right)\) |
| \(\chi_{1003}(378,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{51}{116}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{27}{116}\right)\) | \(e\left(\frac{99}{116}\right)\) | \(e\left(\frac{81}{116}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{75}{116}\right)\) | \(e\left(\frac{11}{116}\right)\) |
| \(\chi_{1003}(387,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{105}{116}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{1}{116}\right)\) | \(e\left(\frac{81}{116}\right)\) | \(e\left(\frac{3}{116}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{47}{58}\right)\) | \(e\left(\frac{93}{116}\right)\) | \(e\left(\frac{9}{116}\right)\) |
| \(\chi_{1003}(404,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{53}{116}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{69}{116}\right)\) | \(e\left(\frac{21}{116}\right)\) | \(e\left(\frac{91}{116}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{37}{116}\right)\) | \(e\left(\frac{41}{116}\right)\) |
| \(\chi_{1003}(421,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{97}{116}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{65}{116}\right)\) | \(e\left(\frac{45}{116}\right)\) | \(e\left(\frac{79}{116}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{13}{116}\right)\) | \(e\left(\frac{5}{116}\right)\) |
| \(\chi_{1003}(446,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{47}{116}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{59}{116}\right)\) | \(e\left(\frac{23}{116}\right)\) | \(e\left(\frac{61}{116}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{47}{58}\right)\) | \(e\left(\frac{35}{116}\right)\) | \(e\left(\frac{67}{116}\right)\) |
| \(\chi_{1003}(455,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{85}{116}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{45}{116}\right)\) | \(e\left(\frac{49}{116}\right)\) | \(e\left(\frac{19}{116}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{9}{116}\right)\) | \(e\left(\frac{57}{116}\right)\) |