from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8016, base_ring=CyclotomicField(166))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,70]))
chi.galois_orbit()
[g,chi] = znchar(Mod(49,8016))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8016\) | |
| 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: | no | |
| 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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8016}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{83}\right)\) | \(e\left(\frac{63}{83}\right)\) | \(e\left(\frac{67}{83}\right)\) | \(e\left(\frac{36}{83}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{62}{83}\right)\) | \(e\left(\frac{70}{83}\right)\) | \(e\left(\frac{21}{83}\right)\) | \(e\left(\frac{79}{83}\right)\) |
| \(\chi_{8016}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{13}{83}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{27}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{17}{83}\right)\) |
| \(\chi_{8016}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{73}{83}\right)\) | \(e\left(\frac{64}{83}\right)\) | \(e\left(\frac{70}{83}\right)\) | \(e\left(\frac{3}{83}\right)\) | \(e\left(\frac{18}{83}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{39}{83}\right)\) |
| \(\chi_{8016}(337,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{83}\right)\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{71}{83}\right)\) | \(e\left(\frac{27}{83}\right)\) | \(e\left(\frac{1}{83}\right)\) | \(e\left(\frac{70}{83}\right)\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{78}{83}\right)\) | \(e\left(\frac{80}{83}\right)\) |
| \(\chi_{8016}(433,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{45}{83}\right)\) | \(e\left(\frac{36}{83}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{80}{83}\right)\) | \(e\left(\frac{39}{83}\right)\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{50}{83}\right)\) | \(e\left(\frac{15}{83}\right)\) | \(e\left(\frac{9}{83}\right)\) |
| \(\chi_{8016}(481,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{82}{83}\right)\) | \(e\left(\frac{48}{83}\right)\) | \(e\left(\frac{55}{83}\right)\) | \(e\left(\frac{63}{83}\right)\) | \(e\left(\frac{30}{83}\right)\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{67}{83}\right)\) | \(e\left(\frac{81}{83}\right)\) | \(e\left(\frac{16}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) |
| \(\chi_{8016}(529,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{83}\right)\) | \(e\left(\frac{62}{83}\right)\) | \(e\left(\frac{33}{83}\right)\) | \(e\left(\frac{71}{83}\right)\) | \(e\left(\frac{18}{83}\right)\) | \(e\left(\frac{15}{83}\right)\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{32}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{29}{83}\right)\) |
| \(\chi_{8016}(577,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{83}\right)\) | \(e\left(\frac{8}{83}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{52}{83}\right)\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{18}{83}\right)\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{55}{83}\right)\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{68}{83}\right)\) |
| \(\chi_{8016}(625,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{70}{83}\right)\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{40}{83}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{33}{83}\right)\) | \(e\left(\frac{32}{83}\right)\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{51}{83}\right)\) | \(e\left(\frac{14}{83}\right)\) |
| \(\chi_{8016}(961,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{79}{83}\right)\) | \(e\left(\frac{30}{83}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{39}{83}\right)\) | \(e\left(\frac{74}{83}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{14}{83}\right)\) | \(e\left(\frac{54}{83}\right)\) | \(e\left(\frac{49}{83}\right)\) |
| \(\chi_{8016}(1009,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{83}\right)\) | \(e\left(\frac{73}{83}\right)\) | \(e\left(\frac{75}{83}\right)\) | \(e\left(\frac{18}{83}\right)\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{35}{83}\right)\) | \(e\left(\frac{52}{83}\right)\) | \(e\left(\frac{81}{83}\right)\) |
| \(\chi_{8016}(1201,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{14}{83}\right)\) | \(e\left(\frac{61}{83}\right)\) | \(e\left(\frac{8}{83}\right)\) | \(e\left(\frac{71}{83}\right)\) | \(e\left(\frac{73}{83}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{34}{83}\right)\) | \(e\left(\frac{60}{83}\right)\) | \(e\left(\frac{36}{83}\right)\) |
| \(\chi_{8016}(1297,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{3}{83}\right)\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{61}{83}\right)\) | \(e\left(\frac{33}{83}\right)\) | \(e\left(\frac{69}{83}\right)\) | \(e\left(\frac{82}{83}\right)\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{1}{83}\right)\) | \(e\left(\frac{67}{83}\right)\) |
| \(\chi_{8016}(1345,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{59}{83}\right)\) | \(e\left(\frac{54}{83}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{10}{83}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{73}{83}\right)\) | \(e\left(\frac{77}{83}\right)\) |
| \(\chi_{8016}(1393,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{26}{83}\right)\) | \(e\left(\frac{44}{83}\right)\) | \(e\left(\frac{9}{83}\right)\) | \(e\left(\frac{54}{83}\right)\) | \(e\left(\frac{69}{83}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{34}{83}\right)\) |
| \(\chi_{8016}(1633,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{72}{83}\right)\) | \(e\left(\frac{30}{83}\right)\) | \(e\left(\frac{24}{83}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{81}{83}\right)\) | \(e\left(\frac{26}{83}\right)\) | \(e\left(\frac{73}{83}\right)\) | \(e\left(\frac{61}{83}\right)\) | \(e\left(\frac{10}{83}\right)\) | \(e\left(\frac{6}{83}\right)\) |
| \(\chi_{8016}(1681,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{83}\right)\) | \(e\left(\frac{75}{83}\right)\) | \(e\left(\frac{60}{83}\right)\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{78}{83}\right)\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{15}{83}\right)\) |
| \(\chi_{8016}(1777,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{33}{83}\right)\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{12}{83}\right)\) | \(e\left(\frac{72}{83}\right)\) | \(e\left(\frac{9}{83}\right)\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{73}{83}\right)\) |
| \(\chi_{8016}(1873,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{83}\right)\) | \(e\left(\frac{42}{83}\right)\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{24}{83}\right)\) | \(e\left(\frac{47}{83}\right)\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{69}{83}\right)\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{14}{83}\right)\) | \(e\left(\frac{25}{83}\right)\) |
| \(\chi_{8016}(1921,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{83}\right)\) | \(e\left(\frac{47}{83}\right)\) | \(e\left(\frac{21}{83}\right)\) | \(e\left(\frac{15}{83}\right)\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{12}{83}\right)\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{71}{83}\right)\) | \(e\left(\frac{26}{83}\right)\) |
| \(\chi_{8016}(1969,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{83}\right)\) | \(e\left(\frac{49}{83}\right)\) | \(e\left(\frac{6}{83}\right)\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{41}{83}\right)\) | \(e\left(\frac{48}{83}\right)\) | \(e\left(\frac{39}{83}\right)\) | \(e\left(\frac{36}{83}\right)\) | \(e\left(\frac{44}{83}\right)\) | \(e\left(\frac{43}{83}\right)\) |
| \(\chi_{8016}(2065,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{79}{83}\right)\) | \(e\left(\frac{6}{83}\right)\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{30}{83}\right)\) | \(e\left(\frac{66}{83}\right)\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{65}{83}\right)\) |
| \(\chi_{8016}(2161,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{83}\right)\) | \(e\left(\frac{12}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{78}{83}\right)\) | \(e\left(\frac{49}{83}\right)\) | \(e\left(\frac{27}{83}\right)\) | \(e\left(\frac{79}{83}\right)\) | \(e\left(\frac{41}{83}\right)\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{19}{83}\right)\) |
| \(\chi_{8016}(2209,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{83}\right)\) | \(e\left(\frac{55}{83}\right)\) | \(e\left(\frac{44}{83}\right)\) | \(e\left(\frac{67}{83}\right)\) | \(e\left(\frac{24}{83}\right)\) | \(e\left(\frac{20}{83}\right)\) | \(e\left(\frac{37}{83}\right)\) | \(e\left(\frac{15}{83}\right)\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{11}{83}\right)\) |
| \(\chi_{8016}(2401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{70}{83}\right)\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{51}{83}\right)\) | \(e\left(\frac{72}{83}\right)\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{41}{83}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{42}{83}\right)\) | \(e\left(\frac{75}{83}\right)\) |
| \(\chi_{8016}(2593,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{74}{83}\right)\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{80}{83}\right)\) | \(e\left(\frac{69}{83}\right)\) | \(e\left(\frac{21}{83}\right)\) | \(e\left(\frac{59}{83}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{61}{83}\right)\) | \(e\left(\frac{20}{83}\right)\) |
| \(\chi_{8016}(2737,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{52}{83}\right)\) | \(e\left(\frac{77}{83}\right)\) | \(e\left(\frac{45}{83}\right)\) | \(e\left(\frac{44}{83}\right)\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{21}{83}\right)\) | \(e\left(\frac{81}{83}\right)\) | \(e\left(\frac{32}{83}\right)\) |
| \(\chi_{8016}(2881,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{42}{83}\right)\) | \(e\left(\frac{30}{83}\right)\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{24}{83}\right)\) | \(e\left(\frac{3}{83}\right)\) | \(e\left(\frac{59}{83}\right)\) | \(e\left(\frac{52}{83}\right)\) |
| \(\chi_{8016}(3025,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{82}{83}\right)\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{49}{83}\right)\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{34}{83}\right)\) | \(e\left(\frac{37}{83}\right)\) |
| \(\chi_{8016}(3121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{83}\right)\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{72}{83}\right)\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{77}{83}\right)\) | \(e\left(\frac{78}{83}\right)\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{30}{83}\right)\) | \(e\left(\frac{18}{83}\right)\) |
| \(\chi_{8016}(3217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{54}{83}\right)\) | \(e\left(\frac{64}{83}\right)\) | \(e\left(\frac{18}{83}\right)\) | \(e\left(\frac{1}{83}\right)\) | \(e\left(\frac{40}{83}\right)\) | \(e\left(\frac{61}{83}\right)\) | \(e\left(\frac{34}{83}\right)\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{49}{83}\right)\) | \(e\left(\frac{46}{83}\right)\) |