from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(668, base_ring=CyclotomicField(166))
M = H._module
chi = DirichletCharacter(H, M([0,22]))
chi.galois_orbit()
[g,chi] = znchar(Mod(9,668))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(668\) | |
| Conductor: | \(167\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(83\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 167.c | 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: | $\Q(\zeta_{83})$ |
| Fixed field: | Number field defined by a degree 83 polynomial |
First 31 of 82 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{668}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{59}{83}\right)\) | \(e\left(\frac{54}{83}\right)\) | \(e\left(\frac{49}{83}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{8}{83}\right)\) |
| \(\chi_{668}(21,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{8}{83}\right)\) | \(e\left(\frac{63}{83}\right)\) | \(e\left(\frac{45}{83}\right)\) | \(e\left(\frac{27}{83}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{6}{83}\right)\) | \(e\left(\frac{62}{83}\right)\) |
| \(\chi_{668}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{1}{83}\right)\) | \(e\left(\frac{35}{83}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{20}{83}\right)\) | \(e\left(\frac{12}{83}\right)\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{46}{83}\right)\) |
| \(\chi_{668}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{78}{83}\right)\) | \(e\left(\frac{75}{83}\right)\) | \(e\left(\frac{52}{83}\right)\) | \(e\left(\frac{73}{83}\right)\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{6}{83}\right)\) | \(e\left(\frac{70}{83}\right)\) | \(e\left(\frac{74}{83}\right)\) | \(e\left(\frac{34}{83}\right)\) | \(e\left(\frac{47}{83}\right)\) |
| \(\chi_{668}(33,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{61}{83}\right)\) | \(e\left(\frac{60}{83}\right)\) | \(e\left(\frac{14}{83}\right)\) | \(e\left(\frac{48}{83}\right)\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{79}{83}\right)\) | \(e\left(\frac{52}{83}\right)\) | \(e\left(\frac{67}{83}\right)\) |
| \(\chi_{668}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{35}{83}\right)\) | \(e\left(\frac{63}{83}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{67}{83}\right)\) | \(e\left(\frac{36}{83}\right)\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{33}{83}\right)\) |
| \(\chi_{668}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{12}{83}\right)\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{26}{83}\right)\) | \(e\left(\frac{82}{83}\right)\) | \(e\left(\frac{44}{83}\right)\) | \(e\left(\frac{9}{83}\right)\) | \(e\left(\frac{10}{83}\right)\) |
| \(\chi_{668}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{33}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{62}{83}\right)\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{79}{83}\right)\) | \(e\left(\frac{64}{83}\right)\) | \(e\left(\frac{6}{83}\right)\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{24}{83}\right)\) |
| \(\chi_{668}(65,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{74}{83}\right)\) | \(e\left(\frac{52}{83}\right)\) | \(e\left(\frac{77}{83}\right)\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{45}{83}\right)\) | \(e\left(\frac{44}{83}\right)\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{68}{83}\right)\) |
| \(\chi_{668}(77,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{73}{83}\right)\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{52}{83}\right)\) | \(e\left(\frac{49}{83}\right)\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{51}{83}\right)\) | \(e\left(\frac{1}{83}\right)\) | \(e\left(\frac{38}{83}\right)\) |
| \(\chi_{668}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{69}{83}\right)\) | \(e\left(\frac{35}{83}\right)\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{15}{83}\right)\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{16}{83}\right)\) |
| \(\chi_{668}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{48}{83}\right)\) | \(e\left(\frac{27}{83}\right)\) | \(e\left(\frac{32}{83}\right)\) | \(e\left(\frac{13}{83}\right)\) | \(e\left(\frac{9}{83}\right)\) | \(e\left(\frac{42}{83}\right)\) | \(e\left(\frac{75}{83}\right)\) | \(e\left(\frac{20}{83}\right)\) | \(e\left(\frac{72}{83}\right)\) | \(e\left(\frac{80}{83}\right)\) |
| \(\chi_{668}(89,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{77}{83}\right)\) | \(e\left(\frac{39}{83}\right)\) | \(e\left(\frac{34}{83}\right)\) | \(e\left(\frac{81}{83}\right)\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{14}{83}\right)\) | \(e\left(\frac{67}{83}\right)\) | \(e\left(\frac{56}{83}\right)\) |
| \(\chi_{668}(93,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{83}\right)\) | \(e\left(\frac{9}{83}\right)\) | \(e\left(\frac{66}{83}\right)\) | \(e\left(\frac{32}{83}\right)\) | \(e\left(\frac{3}{83}\right)\) | \(e\left(\frac{14}{83}\right)\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{62}{83}\right)\) | \(e\left(\frac{24}{83}\right)\) | \(e\left(\frac{82}{83}\right)\) |
| \(\chi_{668}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{83}\right)\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{6}{83}\right)\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{13}{83}\right)\) | \(e\left(\frac{41}{83}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{5}{83}\right)\) |
| \(\chi_{668}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{83}\right)\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{67}{83}\right)\) | \(e\left(\frac{35}{83}\right)\) | \(e\left(\frac{37}{83}\right)\) | \(e\left(\frac{62}{83}\right)\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{73}{83}\right)\) | \(e\left(\frac{47}{83}\right)\) | \(e\left(\frac{43}{83}\right)\) |
| \(\chi_{668}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{83}\right)\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{9}{83}\right)\) | \(e\left(\frac{27}{83}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{60}{83}\right)\) | \(e\left(\frac{16}{83}\right)\) | \(e\left(\frac{41}{83}\right)\) | \(e\left(\frac{64}{83}\right)\) |
| \(\chi_{668}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{83}\right)\) | \(e\left(\frac{26}{83}\right)\) | \(e\left(\frac{80}{83}\right)\) | \(e\left(\frac{74}{83}\right)\) | \(e\left(\frac{64}{83}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{63}{83}\right)\) | \(e\left(\frac{50}{83}\right)\) | \(e\left(\frac{14}{83}\right)\) | \(e\left(\frac{34}{83}\right)\) |
| \(\chi_{668}(141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{83}\right)\) | \(e\left(\frac{30}{83}\right)\) | \(e\left(\frac{54}{83}\right)\) | \(e\left(\frac{79}{83}\right)\) | \(e\left(\frac{10}{83}\right)\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{13}{83}\right)\) | \(e\left(\frac{80}{83}\right)\) | \(e\left(\frac{52}{83}\right)\) |
| \(\chi_{668}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{83}\right)\) | \(e\left(\frac{62}{83}\right)\) | \(e\left(\frac{12}{83}\right)\) | \(e\left(\frac{36}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{78}{83}\right)\) | \(e\left(\frac{80}{83}\right)\) | \(e\left(\frac{49}{83}\right)\) | \(e\left(\frac{27}{83}\right)\) | \(e\left(\frac{30}{83}\right)\) |
| \(\chi_{668}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{54}{83}\right)\) | \(e\left(\frac{20}{83}\right)\) | \(e\left(\frac{36}{83}\right)\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{62}{83}\right)\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{74}{83}\right)\) | \(e\left(\frac{64}{83}\right)\) | \(e\left(\frac{81}{83}\right)\) | \(e\left(\frac{7}{83}\right)\) |
| \(\chi_{668}(173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{83}\right)\) | \(e\left(\frac{67}{83}\right)\) | \(e\left(\frac{21}{83}\right)\) | \(e\left(\frac{63}{83}\right)\) | \(e\left(\frac{50}{83}\right)\) | \(e\left(\frac{12}{83}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{11}{83}\right)\) |
| \(\chi_{668}(181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{83}\right)\) | \(e\left(\frac{79}{83}\right)\) | \(e\left(\frac{26}{83}\right)\) | \(e\left(\frac{78}{83}\right)\) | \(e\left(\frac{54}{83}\right)\) | \(e\left(\frac{3}{83}\right)\) | \(e\left(\frac{35}{83}\right)\) | \(e\left(\frac{37}{83}\right)\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{65}{83}\right)\) |
| \(\chi_{668}(185,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{83}\right)\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{6}{83}\right)\) | \(e\left(\frac{18}{83}\right)\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{39}{83}\right)\) | \(e\left(\frac{40}{83}\right)\) | \(e\left(\frac{66}{83}\right)\) | \(e\left(\frac{55}{83}\right)\) | \(e\left(\frac{15}{83}\right)\) |
| \(\chi_{668}(189,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{42}{83}\right)\) | \(e\left(\frac{34}{83}\right)\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{1}{83}\right)\) | \(e\left(\frac{39}{83}\right)\) | \(e\left(\frac{16}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{59}{83}\right)\) | \(e\left(\frac{63}{83}\right)\) | \(e\left(\frac{70}{83}\right)\) |
| \(\chi_{668}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{83}\right)\) | \(e\left(\frac{49}{83}\right)\) | \(e\left(\frac{55}{83}\right)\) | \(e\left(\frac{82}{83}\right)\) | \(e\left(\frac{44}{83}\right)\) | \(e\left(\frac{67}{83}\right)\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{24}{83}\right)\) | \(e\left(\frac{20}{83}\right)\) | \(e\left(\frac{13}{83}\right)\) |
| \(\chi_{668}(209,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{33}{83}\right)\) | \(e\left(\frac{42}{83}\right)\) | \(e\left(\frac{30}{83}\right)\) | \(e\left(\frac{18}{83}\right)\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{69}{83}\right)\) |
| \(\chi_{668}(217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{21}{83}\right)\) | \(e\left(\frac{71}{83}\right)\) | \(e\left(\frac{47}{83}\right)\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{3}{83}\right)\) | \(e\left(\frac{34}{83}\right)\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{53}{83}\right)\) |
| \(\chi_{668}(221,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{78}{83}\right)\) | \(e\left(\frac{74}{83}\right)\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{26}{83}\right)\) | \(e\left(\frac{66}{83}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{67}{83}\right)\) | \(e\left(\frac{42}{83}\right)\) | \(e\left(\frac{19}{83}\right)\) |
| \(\chi_{668}(225,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{83}\right)\) | \(e\left(\frac{12}{83}\right)\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{15}{83}\right)\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{74}{83}\right)\) | \(e\left(\frac{61}{83}\right)\) | \(e\left(\frac{55}{83}\right)\) | \(e\left(\frac{32}{83}\right)\) | \(e\left(\frac{54}{83}\right)\) |
| \(\chi_{668}(229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{83}\right)\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{34}{83}\right)\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{77}{83}\right)\) | \(e\left(\frac{55}{83}\right)\) | \(e\left(\frac{33}{83}\right)\) | \(e\left(\frac{42}{83}\right)\) | \(e\left(\frac{35}{83}\right)\) | \(e\left(\frac{2}{83}\right)\) |