from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(683, base_ring=CyclotomicField(682))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,683))
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: | \(682\) | 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: | $\Q(\zeta_{341})$ |
| Fixed field: | Number field defined by a degree 682 polynomial (not computed) |
First 31 of 300 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{683}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{15}{31}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{1}{682}\right)\) | \(e\left(\frac{547}{682}\right)\) | \(e\left(\frac{265}{682}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{30}{31}\right)\) | \(e\left(\frac{109}{341}\right)\) | \(e\left(\frac{657}{682}\right)\) |
| \(\chi_{683}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{21}{31}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{547}{682}\right)\) | \(e\left(\frac{493}{682}\right)\) | \(e\left(\frac{371}{682}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{11}{31}\right)\) | \(e\left(\frac{289}{341}\right)\) | \(e\left(\frac{647}{682}\right)\) |
| \(\chi_{683}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{7}{31}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{265}{682}\right)\) | \(e\left(\frac{371}{682}\right)\) | \(e\left(\frac{661}{682}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{14}{31}\right)\) | \(e\left(\frac{241}{341}\right)\) | \(e\left(\frac{195}{682}\right)\) |
| \(\chi_{683}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{28}{31}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{657}{682}\right)\) | \(e\left(\frac{647}{682}\right)\) | \(e\left(\frac{195}{682}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{25}{31}\right)\) | \(e\left(\frac{3}{341}\right)\) | \(e\left(\frac{625}{682}\right)\) |
| \(\chi_{683}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{19}{31}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{427}{682}\right)\) | \(e\left(\frac{325}{682}\right)\) | \(e\left(\frac{625}{682}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{7}{31}\right)\) | \(e\left(\frac{167}{341}\right)\) | \(e\left(\frac{237}{682}\right)\) |
| \(\chi_{683}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{5}{31}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{331}{682}\right)\) | \(e\left(\frac{327}{682}\right)\) | \(e\left(\frac{419}{682}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{10}{31}\right)\) | \(e\left(\frac{274}{341}\right)\) | \(e\left(\frac{591}{682}\right)\) |
| \(\chi_{683}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{6}{31}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{19}{682}\right)\) | \(e\left(\frac{163}{682}\right)\) | \(e\left(\frac{261}{682}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{12}{31}\right)\) | \(e\left(\frac{25}{341}\right)\) | \(e\left(\frac{207}{682}\right)\) |
| \(\chi_{683}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{11}{31}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{195}{682}\right)\) | \(e\left(\frac{273}{682}\right)\) | \(e\left(\frac{525}{682}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{22}{31}\right)\) | \(e\left(\frac{113}{341}\right)\) | \(e\left(\frac{581}{682}\right)\) |
| \(\chi_{683}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{15}{31}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{435}{682}\right)\) | \(e\left(\frac{609}{682}\right)\) | \(e\left(\frac{17}{682}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{30}{31}\right)\) | \(e\left(\frac{16}{341}\right)\) | \(e\left(\frac{37}{682}\right)\) |
| \(\chi_{683}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{28}{31}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{595}{682}\right)\) | \(e\left(\frac{151}{682}\right)\) | \(e\left(\frac{133}{682}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{25}{31}\right)\) | \(e\left(\frac{65}{341}\right)\) | \(e\left(\frac{129}{682}\right)\) |
| \(\chi_{683}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{2}{31}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{399}{682}\right)\) | \(e\left(\frac{13}{682}\right)\) | \(e\left(\frac{25}{682}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{4}{31}\right)\) | \(e\left(\frac{184}{341}\right)\) | \(e\left(\frac{255}{682}\right)\) |
| \(\chi_{683}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{21}{31}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{299}{682}\right)\) | \(e\left(\frac{555}{682}\right)\) | \(e\left(\frac{123}{682}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{11}{31}\right)\) | \(e\left(\frac{196}{341}\right)\) | \(e\left(\frac{27}{682}\right)\) |
| \(\chi_{683}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{7}{31}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{17}{682}\right)\) | \(e\left(\frac{433}{682}\right)\) | \(e\left(\frac{413}{682}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{14}{31}\right)\) | \(e\left(\frac{148}{341}\right)\) | \(e\left(\frac{257}{682}\right)\) |
| \(\chi_{683}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{3}{31}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{149}{682}\right)\) | \(e\left(\frac{345}{682}\right)\) | \(e\left(\frac{611}{682}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{6}{31}\right)\) | \(e\left(\frac{214}{341}\right)\) | \(e\left(\frac{367}{682}\right)\) |
| \(\chi_{683}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{18}{31}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{305}{682}\right)\) | \(e\left(\frac{427}{682}\right)\) | \(e\left(\frac{349}{682}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{5}{31}\right)\) | \(e\left(\frac{168}{341}\right)\) | \(e\left(\frac{559}{682}\right)\) |
| \(\chi_{683}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{20}{31}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{487}{682}\right)\) | \(e\left(\frac{409}{682}\right)\) | \(e\left(\frac{157}{682}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{9}{31}\right)\) | \(e\left(\frac{228}{341}\right)\) | \(e\left(\frac{101}{682}\right)\) |
| \(\chi_{683}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{9}{31}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{75}{682}\right)\) | \(e\left(\frac{105}{682}\right)\) | \(e\left(\frac{97}{682}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{18}{31}\right)\) | \(e\left(\frac{332}{341}\right)\) | \(e\left(\frac{171}{682}\right)\) |
| \(\chi_{683}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{1}{31}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{463}{682}\right)\) | \(e\left(\frac{239}{682}\right)\) | \(e\left(\frac{617}{682}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{2}{31}\right)\) | \(e\left(\frac{340}{341}\right)\) | \(e\left(\frac{19}{682}\right)\) |
| \(\chi_{683}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{15}{31}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{497}{682}\right)\) | \(e\left(\frac{423}{682}\right)\) | \(e\left(\frac{79}{682}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{30}{31}\right)\) | \(e\left(\frac{295}{341}\right)\) | \(e\left(\frac{533}{682}\right)\) |
| \(\chi_{683}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{28}{31}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{409}{682}\right)\) | \(e\left(\frac{27}{682}\right)\) | \(e\left(\frac{629}{682}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{25}{31}\right)\) | \(e\left(\frac{251}{341}\right)\) | \(e\left(\frac{5}{682}\right)\) |
| \(\chi_{683}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{26}{31}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{661}{682}\right)\) | \(e\left(\frac{107}{682}\right)\) | \(e\left(\frac{573}{682}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{21}{31}\right)\) | \(e\left(\frac{98}{341}\right)\) | \(e\left(\frac{525}{682}\right)\) |
| \(\chi_{683}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{23}{31}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{543}{682}\right)\) | \(e\left(\frac{351}{682}\right)\) | \(e\left(\frac{675}{682}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{15}{31}\right)\) | \(e\left(\frac{194}{341}\right)\) | \(e\left(\frac{65}{682}\right)\) |
| \(\chi_{683}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{30}{31}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{219}{682}\right)\) | \(e\left(\frac{443}{682}\right)\) | \(e\left(\frac{65}{682}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{29}{31}\right)\) | \(e\left(\frac{1}{341}\right)\) | \(e\left(\frac{663}{682}\right)\) |
| \(\chi_{683}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{27}{31}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{349}{682}\right)\) | \(e\left(\frac{625}{682}\right)\) | \(e\left(\frac{415}{682}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{23}{31}\right)\) | \(e\left(\frac{190}{341}\right)\) | \(e\left(\frac{141}{682}\right)\) |
| \(\chi_{683}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{19}{31}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{179}{682}\right)\) | \(e\left(\frac{387}{682}\right)\) | \(e\left(\frac{377}{682}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{7}{31}\right)\) | \(e\left(\frac{74}{341}\right)\) | \(e\left(\frac{299}{682}\right)\) |
| \(\chi_{683}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{1}{31}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{525}{682}\right)\) | \(e\left(\frac{53}{682}\right)\) | \(e\left(\frac{679}{682}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{2}{31}\right)\) | \(e\left(\frac{278}{341}\right)\) | \(e\left(\frac{515}{682}\right)\) |
| \(\chi_{683}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{29}{31}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{345}{682}\right)\) | \(e\left(\frac{483}{682}\right)\) | \(e\left(\frac{37}{682}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{27}{31}\right)\) | \(e\left(\frac{95}{341}\right)\) | \(e\left(\frac{241}{682}\right)\) |
| \(\chi_{683}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{5}{31}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{83}{682}\right)\) | \(e\left(\frac{389}{682}\right)\) | \(e\left(\frac{171}{682}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{10}{31}\right)\) | \(e\left(\frac{181}{341}\right)\) | \(e\left(\frac{653}{682}\right)\) |
| \(\chi_{683}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{18}{31}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{243}{682}\right)\) | \(e\left(\frac{613}{682}\right)\) | \(e\left(\frac{287}{682}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{5}{31}\right)\) | \(e\left(\frac{230}{341}\right)\) | \(e\left(\frac{63}{682}\right)\) |
| \(\chi_{683}(68,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{6}{31}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{453}{682}\right)\) | \(e\left(\frac{225}{682}\right)\) | \(e\left(\frac{13}{682}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{12}{31}\right)\) | \(e\left(\frac{273}{341}\right)\) | \(e\left(\frac{269}{682}\right)\) |
| \(\chi_{683}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{23}{31}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{47}{682}\right)\) | \(e\left(\frac{475}{682}\right)\) | \(e\left(\frac{179}{682}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{15}{31}\right)\) | \(e\left(\frac{8}{341}\right)\) | \(e\left(\frac{189}{682}\right)\) |