from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3006, base_ring=CyclotomicField(166))
M = H._module
chi = DirichletCharacter(H, M([0,58]))
chi.galois_orbit()
[g,chi] = znchar(Mod(19,3006))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3006\) | |
| 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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3006}(19,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{82}{83}\right)\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{49}{83}\right)\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{34}{83}\right)\) | \(e\left(\frac{37}{83}\right)\) |
| \(\chi_{3006}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{1}{83}\right)\) | \(e\left(\frac{34}{83}\right)\) | \(e\left(\frac{48}{83}\right)\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{55}{83}\right)\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{50}{83}\right)\) |
| \(\chi_{3006}(181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{83}\right)\) | \(e\left(\frac{26}{83}\right)\) | \(e\left(\frac{54}{83}\right)\) | \(e\left(\frac{3}{83}\right)\) | \(e\left(\frac{37}{83}\right)\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{75}{83}\right)\) | \(e\left(\frac{64}{83}\right)\) | \(e\left(\frac{55}{83}\right)\) |
| \(\chi_{3006}(199,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{14}{83}\right)\) | \(e\left(\frac{61}{83}\right)\) | \(e\left(\frac{8}{83}\right)\) | \(e\left(\frac{71}{83}\right)\) | \(e\left(\frac{73}{83}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{34}{83}\right)\) | \(e\left(\frac{60}{83}\right)\) | \(e\left(\frac{36}{83}\right)\) |
| \(\chi_{3006}(217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{83}\right)\) | \(e\left(\frac{71}{83}\right)\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{34}{83}\right)\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{42}{83}\right)\) | \(e\left(\frac{79}{83}\right)\) | \(e\left(\frac{64}{83}\right)\) |
| \(\chi_{3006}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{73}{83}\right)\) | \(e\left(\frac{64}{83}\right)\) | \(e\left(\frac{70}{83}\right)\) | \(e\left(\frac{3}{83}\right)\) | \(e\left(\frac{18}{83}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{39}{83}\right)\) |
| \(\chi_{3006}(343,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{59}{83}\right)\) | \(e\left(\frac{54}{83}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{10}{83}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{73}{83}\right)\) | \(e\left(\frac{77}{83}\right)\) |
| \(\chi_{3006}(361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{47}{83}\right)\) | \(e\left(\frac{81}{83}\right)\) | \(e\left(\frac{3}{83}\right)\) | \(e\left(\frac{44}{83}\right)\) | \(e\left(\frac{15}{83}\right)\) | \(e\left(\frac{33}{83}\right)\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{74}{83}\right)\) |
| \(\chi_{3006}(397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{70}{83}\right)\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{51}{83}\right)\) | \(e\left(\frac{72}{83}\right)\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{41}{83}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{42}{83}\right)\) | \(e\left(\frac{75}{83}\right)\) |
| \(\chi_{3006}(415,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{35}{83}\right)\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{20}{83}\right)\) | \(e\left(\frac{44}{83}\right)\) | \(e\left(\frac{63}{83}\right)\) | \(e\left(\frac{71}{83}\right)\) |
| \(\chi_{3006}(433,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{45}{83}\right)\) | \(e\left(\frac{36}{83}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{80}{83}\right)\) | \(e\left(\frac{39}{83}\right)\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{50}{83}\right)\) | \(e\left(\frac{15}{83}\right)\) | \(e\left(\frac{9}{83}\right)\) |
| \(\chi_{3006}(505,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{83}\right)\) | \(e\left(\frac{72}{83}\right)\) | \(e\left(\frac{41}{83}\right)\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{45}{83}\right)\) | \(e\left(\frac{79}{83}\right)\) | \(e\left(\frac{59}{83}\right)\) | \(e\left(\frac{80}{83}\right)\) | \(e\left(\frac{24}{83}\right)\) | \(e\left(\frac{31}{83}\right)\) |
| \(\chi_{3006}(523,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{83}\right)\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{39}{83}\right)\) | \(e\left(\frac{16}{83}\right)\) | \(e\left(\frac{59}{83}\right)\) | \(e\left(\frac{63}{83}\right)\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{37}{83}\right)\) | \(e\left(\frac{72}{83}\right)\) |
| \(\chi_{3006}(559,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{83}\right)\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{74}{83}\right)\) | \(e\left(\frac{55}{83}\right)\) | \(e\left(\frac{32}{83}\right)\) | \(e\left(\frac{26}{83}\right)\) | \(e\left(\frac{24}{83}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{1}{83}\right)\) |
| \(\chi_{3006}(577,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{83}\right)\) | \(e\left(\frac{8}{83}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{52}{83}\right)\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{18}{83}\right)\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{55}{83}\right)\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{68}{83}\right)\) |
| \(\chi_{3006}(595,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{83}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{1}{83}\right)\) | \(e\left(\frac{60}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{8}{83}\right)\) | \(e\left(\frac{48}{83}\right)\) | \(e\left(\frac{6}{83}\right)\) | \(e\left(\frac{35}{83}\right)\) | \(e\left(\frac{21}{83}\right)\) |
| \(\chi_{3006}(613,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{51}{83}\right)\) | \(e\left(\frac{74}{83}\right)\) | \(e\left(\frac{41}{83}\right)\) | \(e\left(\frac{63}{83}\right)\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{66}{83}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{60}{83}\right)\) |
| \(\chi_{3006}(631,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{72}{83}\right)\) | \(e\left(\frac{30}{83}\right)\) | \(e\left(\frac{24}{83}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{81}{83}\right)\) | \(e\left(\frac{26}{83}\right)\) | \(e\left(\frac{73}{83}\right)\) | \(e\left(\frac{61}{83}\right)\) | \(e\left(\frac{10}{83}\right)\) | \(e\left(\frac{6}{83}\right)\) |
| \(\chi_{3006}(757,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{83}\right)\) | \(e\left(\frac{39}{83}\right)\) | \(e\left(\frac{81}{83}\right)\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{14}{83}\right)\) | \(e\left(\frac{67}{83}\right)\) | \(e\left(\frac{70}{83}\right)\) | \(e\left(\frac{71}{83}\right)\) | \(e\left(\frac{13}{83}\right)\) | \(e\left(\frac{41}{83}\right)\) |
| \(\chi_{3006}(775,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{33}{83}\right)\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{12}{83}\right)\) | \(e\left(\frac{72}{83}\right)\) | \(e\left(\frac{9}{83}\right)\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{73}{83}\right)\) |
| \(\chi_{3006}(847,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{80}{83}\right)\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{66}{83}\right)\) | \(e\left(\frac{64}{83}\right)\) | \(e\left(\frac{8}{83}\right)\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{28}{83}\right)\) |
| \(\chi_{3006}(883,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{83}\right)\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{70}{83}\right)\) | \(e\left(\frac{50}{83}\right)\) | \(e\left(\frac{8}{83}\right)\) | \(e\left(\frac{62}{83}\right)\) | \(e\left(\frac{40}{83}\right)\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{59}{83}\right)\) |
| \(\chi_{3006}(901,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{83}\right)\) | \(e\left(\frac{13}{83}\right)\) | \(e\left(\frac{27}{83}\right)\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{60}{83}\right)\) | \(e\left(\frac{50}{83}\right)\) | \(e\left(\frac{51}{83}\right)\) | \(e\left(\frac{79}{83}\right)\) | \(e\left(\frac{32}{83}\right)\) | \(e\left(\frac{69}{83}\right)\) |
| \(\chi_{3006}(919,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{83}\right)\) | \(e\left(\frac{47}{83}\right)\) | \(e\left(\frac{21}{83}\right)\) | \(e\left(\frac{15}{83}\right)\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{12}{83}\right)\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{71}{83}\right)\) | \(e\left(\frac{26}{83}\right)\) |
| \(\chi_{3006}(1009,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{83}\right)\) | \(e\left(\frac{73}{83}\right)\) | \(e\left(\frac{75}{83}\right)\) | \(e\left(\frac{18}{83}\right)\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{35}{83}\right)\) | \(e\left(\frac{52}{83}\right)\) | \(e\left(\frac{81}{83}\right)\) |
| \(\chi_{3006}(1027,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{83}\right)\) | \(e\left(\frac{35}{83}\right)\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{20}{83}\right)\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{16}{83}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{67}{83}\right)\) | \(e\left(\frac{7}{83}\right)\) |
| \(\chi_{3006}(1063,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{79}{83}\right)\) | \(e\left(\frac{6}{83}\right)\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{30}{83}\right)\) | \(e\left(\frac{66}{83}\right)\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{65}{83}\right)\) |
| \(\chi_{3006}(1099,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{13}{83}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{27}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{17}{83}\right)\) |
| \(\chi_{3006}(1117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{83}\right)\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{72}{83}\right)\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{77}{83}\right)\) | \(e\left(\frac{78}{83}\right)\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{30}{83}\right)\) | \(e\left(\frac{18}{83}\right)\) |
| \(\chi_{3006}(1135,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{9}{83}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{16}{83}\right)\) | \(e\left(\frac{41}{83}\right)\) | \(e\left(\frac{80}{83}\right)\) | \(e\left(\frac{10}{83}\right)\) | \(e\left(\frac{3}{83}\right)\) | \(e\left(\frac{35}{83}\right)\) |
| \(\chi_{3006}(1171,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{83}\right)\) | \(e\left(\frac{36}{83}\right)\) | \(e\left(\frac{62}{83}\right)\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{64}{83}\right)\) | \(e\left(\frac{81}{83}\right)\) | \(e\left(\frac{71}{83}\right)\) | \(e\left(\frac{40}{83}\right)\) | \(e\left(\frac{12}{83}\right)\) | \(e\left(\frac{57}{83}\right)\) |