from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(683, base_ring=CyclotomicField(682))
M = H._module
chi = DirichletCharacter(H, M([218]))
chi.galois_orbit()
[g,chi] = znchar(Mod(10,683))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(683\) | |
| Conductor: | \(683\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(341\) | 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: | $\Q(\zeta_{341})$ |
| Fixed field: | Number field defined by a degree 341 polynomial (not computed) |
First 31 of 300 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{683}(10,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{15}{31}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{109}{341}\right)\) | \(e\left(\frac{289}{341}\right)\) | \(e\left(\frac{241}{341}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{30}{31}\right)\) | \(e\left(\frac{233}{341}\right)\) | \(e\left(\frac{3}{341}\right)\) |
| \(\chi_{683}(12,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{21}{31}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{41}{341}\right)\) | \(e\left(\frac{262}{341}\right)\) | \(e\left(\frac{294}{341}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{11}{31}\right)\) | \(e\left(\frac{72}{341}\right)\) | \(e\left(\frac{339}{341}\right)\) |
| \(\chi_{683}(14,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{7}{31}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{241}{341}\right)\) | \(e\left(\frac{201}{341}\right)\) | \(e\left(\frac{98}{341}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{14}{31}\right)\) | \(e\left(\frac{24}{341}\right)\) | \(e\left(\frac{113}{341}\right)\) |
| \(\chi_{683}(19,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{20}{31}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{135}{341}\right)\) | \(e\left(\frac{189}{341}\right)\) | \(e\left(\frac{311}{341}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{9}{31}\right)\) | \(e\left(\frac{104}{341}\right)\) | \(e\left(\frac{35}{341}\right)\) |
| \(\chi_{683}(22,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{28}{31}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{96}{341}\right)\) | \(e\left(\frac{339}{341}\right)\) | \(e\left(\frac{206}{341}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{25}{31}\right)\) | \(e\left(\frac{127}{341}\right)\) | \(e\left(\frac{328}{341}\right)\) |
| \(\chi_{683}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{30}{31}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{341}\right)\) | \(e\left(\frac{206}{341}\right)\) | \(e\left(\frac{265}{341}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{29}{31}\right)\) | \(e\left(\frac{218}{341}\right)\) | \(e\left(\frac{316}{341}\right)\) |
| \(\chi_{683}(26,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{19}{31}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{322}{341}\right)\) | \(e\left(\frac{178}{341}\right)\) | \(e\left(\frac{80}{341}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{7}{31}\right)\) | \(e\left(\frac{291}{341}\right)\) | \(e\left(\frac{134}{341}\right)\) |
| \(\chi_{683}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{29}{31}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{64}{341}\right)\) | \(e\left(\frac{226}{341}\right)\) | \(e\left(\frac{251}{341}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{27}{31}\right)\) | \(e\left(\frac{312}{341}\right)\) | \(e\left(\frac{105}{341}\right)\) |
| \(\chi_{683}(30,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{31}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{274}{341}\right)\) | \(e\left(\frac{179}{341}\right)\) | \(e\left(\frac{318}{341}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{10}{31}\right)\) | \(e\left(\frac{57}{341}\right)\) | \(e\left(\frac{311}{341}\right)\) |
| \(\chi_{683}(34,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{6}{31}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{118}{341}\right)\) | \(e\left(\frac{97}{341}\right)\) | \(e\left(\frac{239}{341}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{12}{31}\right)\) | \(e\left(\frac{149}{341}\right)\) | \(e\left(\frac{119}{341}\right)\) |
| \(\chi_{683}(35,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{22}{31}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{133}{341}\right)\) | \(e\left(\frac{118}{341}\right)\) | \(e\left(\frac{122}{341}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{31}\right)\) | \(e\left(\frac{9}{341}\right)\) | \(e\left(\frac{85}{341}\right)\) |
| \(\chi_{683}(36,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{11}{31}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{206}{341}\right)\) | \(e\left(\frac{152}{341}\right)\) | \(e\left(\frac{30}{341}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{22}{31}\right)\) | \(e\left(\frac{237}{341}\right)\) | \(e\left(\frac{306}{341}\right)\) |
| \(\chi_{683}(40,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{15}{31}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{326}{341}\right)\) | \(e\left(\frac{320}{341}\right)\) | \(e\left(\frac{117}{341}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{30}{31}\right)\) | \(e\left(\frac{140}{341}\right)\) | \(e\left(\frac{34}{341}\right)\) |
| \(\chi_{683}(42,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{28}{31}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{65}{341}\right)\) | \(e\left(\frac{91}{341}\right)\) | \(e\left(\frac{175}{341}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{25}{31}\right)\) | \(e\left(\frac{189}{341}\right)\) | \(e\left(\frac{80}{341}\right)\) |
| \(\chi_{683}(48,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{21}{31}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{258}{341}\right)\) | \(e\left(\frac{293}{341}\right)\) | \(e\left(\frac{170}{341}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{11}{31}\right)\) | \(e\left(\frac{320}{341}\right)\) | \(e\left(\frac{29}{341}\right)\) |
| \(\chi_{683}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{14}{31}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{265}{341}\right)\) | \(e\left(\frac{30}{341}\right)\) | \(e\left(\frac{320}{341}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{28}{31}\right)\) | \(e\left(\frac{141}{341}\right)\) | \(e\left(\frac{195}{341}\right)\) |
| \(\chi_{683}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{2}{31}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{60}{341}\right)\) | \(e\left(\frac{84}{341}\right)\) | \(e\left(\frac{214}{341}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{4}{31}\right)\) | \(e\left(\frac{122}{341}\right)\) | \(e\left(\frac{205}{341}\right)\) |
| \(\chi_{683}(55,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{12}{31}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{329}{341}\right)\) | \(e\left(\frac{256}{341}\right)\) | \(e\left(\frac{230}{341}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{24}{31}\right)\) | \(e\left(\frac{112}{341}\right)\) | \(e\left(\frac{300}{341}\right)\) |
| \(\chi_{683}(56,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{7}{31}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{117}{341}\right)\) | \(e\left(\frac{232}{341}\right)\) | \(e\left(\frac{315}{341}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{14}{31}\right)\) | \(e\left(\frac{272}{341}\right)\) | \(e\left(\frac{144}{341}\right)\) |
| \(\chi_{683}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{10}{31}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{300}{341}\right)\) | \(e\left(\frac{79}{341}\right)\) | \(e\left(\frac{47}{341}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{20}{31}\right)\) | \(e\left(\frac{269}{341}\right)\) | \(e\left(\frac{2}{341}\right)\) |
| \(\chi_{683}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{9}{31}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{301}{341}\right)\) | \(e\left(\frac{285}{341}\right)\) | \(e\left(\frac{312}{341}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{18}{31}\right)\) | \(e\left(\frac{146}{341}\right)\) | \(e\left(\frac{318}{341}\right)\) |
| \(\chi_{683}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{24}{31}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{224}{341}\right)\) | \(e\left(\frac{109}{341}\right)\) | \(e\left(\frac{26}{341}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{17}{31}\right)\) | \(e\left(\frac{69}{341}\right)\) | \(e\left(\frac{197}{341}\right)\) |
| \(\chi_{683}(62,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{3}{31}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{183}{341}\right)\) | \(e\left(\frac{188}{341}\right)\) | \(e\left(\frac{73}{341}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{6}{31}\right)\) | \(e\left(\frac{338}{341}\right)\) | \(e\left(\frac{199}{341}\right)\) |
| \(\chi_{683}(65,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{3}{31}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{214}{341}\right)\) | \(e\left(\frac{95}{341}\right)\) | \(e\left(\frac{104}{341}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{6}{31}\right)\) | \(e\left(\frac{276}{341}\right)\) | \(e\left(\frac{106}{341}\right)\) |
| \(\chi_{683}(66,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{18}{31}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{261}{341}\right)\) | \(e\left(\frac{229}{341}\right)\) | \(e\left(\frac{283}{341}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{31}\right)\) | \(e\left(\frac{292}{341}\right)\) | \(e\left(\frac{295}{341}\right)\) |
| \(\chi_{683}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{17}{31}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{45}{341}\right)\) | \(e\left(\frac{63}{341}\right)\) | \(e\left(\frac{331}{341}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{3}{31}\right)\) | \(e\left(\frac{262}{341}\right)\) | \(e\left(\frac{239}{341}\right)\) |
| \(\chi_{683}(74,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{26}{31}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{191}{341}\right)\) | \(e\left(\frac{131}{341}\right)\) | \(e\left(\frac{147}{341}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{21}{31}\right)\) | \(e\left(\frac{36}{341}\right)\) | \(e\left(\frac{340}{341}\right)\) |
| \(\chi_{683}(75,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{20}{31}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{166}{341}\right)\) | \(e\left(\frac{96}{341}\right)\) | \(e\left(\frac{1}{341}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{9}{31}\right)\) | \(e\left(\frac{42}{341}\right)\) | \(e\left(\frac{283}{341}\right)\) |
| \(\chi_{683}(77,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{4}{31}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{120}{341}\right)\) | \(e\left(\frac{168}{341}\right)\) | \(e\left(\frac{87}{341}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{8}{31}\right)\) | \(e\left(\frac{244}{341}\right)\) | \(e\left(\frac{69}{341}\right)\) |
| \(\chi_{683}(78,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{9}{31}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{146}{341}\right)\) | \(e\left(\frac{68}{341}\right)\) | \(e\left(\frac{157}{341}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{18}{31}\right)\) | \(e\left(\frac{115}{341}\right)\) | \(e\left(\frac{101}{341}\right)\) |
| \(\chi_{683}(82,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{1}{31}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{340}{341}\right)\) | \(e\left(\frac{135}{341}\right)\) | \(e\left(\frac{76}{341}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{2}{31}\right)\) | \(e\left(\frac{123}{341}\right)\) | \(e\left(\frac{25}{341}\right)\) |