from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(167, base_ring=CyclotomicField(166))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,167))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(167\) | |
| Conductor: | \(167\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(166\) | 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: | $\Q(\zeta_{83})$ |
| Fixed field: | Number field defined by a degree 166 polynomial (not computed) |
First 31 of 82 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{167}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{20}{83}\right)\) | \(e\left(\frac{47}{83}\right)\) | \(e\left(\frac{40}{83}\right)\) | \(e\left(\frac{1}{166}\right)\) | \(e\left(\frac{67}{83}\right)\) | \(e\left(\frac{59}{83}\right)\) | \(e\left(\frac{60}{83}\right)\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{41}{166}\right)\) | \(e\left(\frac{14}{83}\right)\) |
| \(\chi_{167}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{83}\right)\) | \(e\left(\frac{18}{83}\right)\) | \(e\left(\frac{63}{83}\right)\) | \(e\left(\frac{41}{166}\right)\) | \(e\left(\frac{8}{83}\right)\) | \(e\left(\frac{12}{83}\right)\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{36}{83}\right)\) | \(e\left(\frac{21}{166}\right)\) | \(e\left(\frac{76}{83}\right)\) |
| \(\chi_{167}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{27}{83}\right)\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{103}{166}\right)\) | \(e\left(\frac{12}{83}\right)\) | \(e\left(\frac{18}{83}\right)\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{54}{83}\right)\) | \(e\left(\frac{73}{166}\right)\) | \(e\left(\frac{31}{83}\right)\) |
| \(\chi_{167}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{74}{83}\right)\) | \(e\left(\frac{66}{83}\right)\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{95}{166}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{44}{83}\right)\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{49}{83}\right)\) | \(e\left(\frac{77}{166}\right)\) | \(e\left(\frac{2}{83}\right)\) |
| \(\chi_{167}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{64}{83}\right)\) | \(e\left(\frac{1}{83}\right)\) | \(e\left(\frac{45}{83}\right)\) | \(e\left(\frac{53}{166}\right)\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{26}{83}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{15}{166}\right)\) | \(e\left(\frac{78}{83}\right)\) |
| \(\chi_{167}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{72}{83}\right)\) | \(e\left(\frac{3}{83}\right)\) | \(e\left(\frac{81}{166}\right)\) | \(e\left(\frac{32}{83}\right)\) | \(e\left(\frac{48}{83}\right)\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{61}{83}\right)\) | \(e\left(\frac{1}{166}\right)\) | \(e\left(\frac{55}{83}\right)\) |
| \(\chi_{167}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{83}\right)\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{59}{83}\right)\) | \(e\left(\frac{99}{166}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{47}{83}\right)\) | \(e\left(\frac{10}{83}\right)\) | \(e\left(\frac{75}{166}\right)\) | \(e\left(\frac{58}{83}\right)\) |
| \(\chi_{167}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{81}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{143}{166}\right)\) | \(e\left(\frac{36}{83}\right)\) | \(e\left(\frac{54}{83}\right)\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{79}{83}\right)\) | \(e\left(\frac{53}{166}\right)\) | \(e\left(\frac{10}{83}\right)\) |
| \(\chi_{167}(30,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{44}{83}\right)\) | \(e\left(\frac{37}{83}\right)\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{135}{166}\right)\) | \(e\left(\frac{81}{83}\right)\) | \(e\left(\frac{80}{83}\right)\) | \(e\left(\frac{49}{83}\right)\) | \(e\left(\frac{74}{83}\right)\) | \(e\left(\frac{57}{166}\right)\) | \(e\left(\frac{64}{83}\right)\) |
| \(\chi_{167}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{34}{83}\right)\) | \(e\left(\frac{55}{83}\right)\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{93}{166}\right)\) | \(e\left(\frac{6}{83}\right)\) | \(e\left(\frac{9}{83}\right)\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{27}{83}\right)\) | \(e\left(\frac{161}{166}\right)\) | \(e\left(\frac{57}{83}\right)\) |
| \(\chi_{167}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{32}{83}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{119}{166}\right)\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{49}{83}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{64}{83}\right)\) | \(e\left(\frac{65}{166}\right)\) | \(e\left(\frac{6}{83}\right)\) |
| \(\chi_{167}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{45}{83}\right)\) | \(e\left(\frac{33}{83}\right)\) | \(e\left(\frac{61}{166}\right)\) | \(e\left(\frac{20}{83}\right)\) | \(e\left(\frac{30}{83}\right)\) | \(e\left(\frac{8}{83}\right)\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{11}{166}\right)\) | \(e\left(\frac{24}{83}\right)\) |
| \(\chi_{167}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{83}\right)\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{78}{83}\right)\) | \(e\left(\frac{31}{166}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{3}{83}\right)\) | \(e\left(\frac{34}{83}\right)\) | \(e\left(\frac{9}{83}\right)\) | \(e\left(\frac{109}{166}\right)\) | \(e\left(\frac{19}{83}\right)\) |
| \(\chi_{167}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{83}\right)\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{26}{83}\right)\) | \(e\left(\frac{121}{166}\right)\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{1}{83}\right)\) | \(e\left(\frac{39}{83}\right)\) | \(e\left(\frac{3}{83}\right)\) | \(e\left(\frac{147}{166}\right)\) | \(e\left(\frac{34}{83}\right)\) |
| \(\chi_{167}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{77}{83}\right)\) | \(e\left(\frac{62}{83}\right)\) | \(e\left(\frac{97}{166}\right)\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{79}{83}\right)\) | \(e\left(\frac{10}{83}\right)\) | \(e\left(\frac{71}{83}\right)\) | \(e\left(\frac{159}{166}\right)\) | \(e\left(\frac{30}{83}\right)\) |
| \(\chi_{167}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{80}{83}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{77}{83}\right)\) | \(e\left(\frac{87}{166}\right)\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{70}{83}\right)\) | \(e\left(\frac{74}{83}\right)\) | \(e\left(\frac{44}{83}\right)\) | \(e\left(\frac{81}{166}\right)\) | \(e\left(\frac{56}{83}\right)\) |
| \(\chi_{167}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{83}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{23}{166}\right)\) | \(e\left(\frac{47}{83}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{52}{83}\right)\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{113}{166}\right)\) | \(e\left(\frac{73}{83}\right)\) |
| \(\chi_{167}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{83}\right)\) | \(e\left(\frac{59}{83}\right)\) | \(e\left(\frac{82}{83}\right)\) | \(e\left(\frac{139}{166}\right)\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{67}{83}\right)\) | \(e\left(\frac{40}{83}\right)\) | \(e\left(\frac{35}{83}\right)\) | \(e\left(\frac{55}{166}\right)\) | \(e\left(\frac{37}{83}\right)\) |
| \(\chi_{167}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{83}\right)\) | \(e\left(\frac{20}{83}\right)\) | \(e\left(\frac{70}{83}\right)\) | \(e\left(\frac{147}{166}\right)\) | \(e\left(\frac{55}{83}\right)\) | \(e\left(\frac{41}{83}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{40}{83}\right)\) | \(e\left(\frac{51}{166}\right)\) | \(e\left(\frac{66}{83}\right)\) |
| \(\chi_{167}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{83}\right)\) | \(e\left(\frac{52}{83}\right)\) | \(e\left(\frac{16}{83}\right)\) | \(e\left(\frac{17}{166}\right)\) | \(e\left(\frac{60}{83}\right)\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{24}{83}\right)\) | \(e\left(\frac{21}{83}\right)\) | \(e\left(\frac{33}{166}\right)\) | \(e\left(\frac{72}{83}\right)\) |
| \(\chi_{167}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{60}{83}\right)\) | \(e\left(\frac{44}{83}\right)\) | \(e\left(\frac{109}{166}\right)\) | \(e\left(\frac{82}{83}\right)\) | \(e\left(\frac{40}{83}\right)\) | \(e\left(\frac{66}{83}\right)\) | \(e\left(\frac{37}{83}\right)\) | \(e\left(\frac{153}{166}\right)\) | \(e\left(\frac{32}{83}\right)\) |
| \(\chi_{167}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{82}{83}\right)\) | \(e\left(\frac{35}{83}\right)\) | \(e\left(\frac{81}{83}\right)\) | \(e\left(\frac{29}{166}\right)\) | \(e\left(\frac{34}{83}\right)\) | \(e\left(\frac{51}{83}\right)\) | \(e\left(\frac{80}{83}\right)\) | \(e\left(\frac{70}{83}\right)\) | \(e\left(\frac{27}{166}\right)\) | \(e\left(\frac{74}{83}\right)\) |
| \(\chi_{167}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{82}{83}\right)\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{113}{166}\right)\) | \(e\left(\frac{18}{83}\right)\) | \(e\left(\frac{27}{83}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{81}{83}\right)\) | \(e\left(\frac{151}{166}\right)\) | \(e\left(\frac{5}{83}\right)\) |
| \(\chi_{167}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{83}\right)\) | \(e\left(\frac{8}{83}\right)\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{9}{166}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{33}{83}\right)\) | \(e\left(\frac{42}{83}\right)\) | \(e\left(\frac{16}{83}\right)\) | \(e\left(\frac{37}{166}\right)\) | \(e\left(\frac{43}{83}\right)\) |
| \(\chi_{167}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{83}\right)\) | \(e\left(\frac{36}{83}\right)\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{165}{166}\right)\) | \(e\left(\frac{16}{83}\right)\) | \(e\left(\frac{24}{83}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{72}{83}\right)\) | \(e\left(\frac{125}{166}\right)\) | \(e\left(\frac{69}{83}\right)\) |
| \(\chi_{167}(68,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{26}{83}\right)\) | \(e\left(\frac{8}{83}\right)\) | \(e\left(\frac{133}{166}\right)\) | \(e\left(\frac{30}{83}\right)\) | \(e\left(\frac{45}{83}\right)\) | \(e\left(\frac{12}{83}\right)\) | \(e\left(\frac{52}{83}\right)\) | \(e\left(\frac{141}{166}\right)\) | \(e\left(\frac{36}{83}\right)\) |
| \(\chi_{167}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{42}{83}\right)\) | \(e\left(\frac{24}{83}\right)\) | \(e\left(\frac{1}{83}\right)\) | \(e\left(\frac{27}{166}\right)\) | \(e\left(\frac{66}{83}\right)\) | \(e\left(\frac{16}{83}\right)\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{48}{83}\right)\) | \(e\left(\frac{111}{166}\right)\) | \(e\left(\frac{46}{83}\right)\) |
| \(\chi_{167}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{26}{83}\right)\) | \(e\left(\frac{3}{83}\right)\) | \(e\left(\frac{52}{83}\right)\) | \(e\left(\frac{159}{166}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{78}{83}\right)\) | \(e\left(\frac{6}{83}\right)\) | \(e\left(\frac{45}{166}\right)\) | \(e\left(\frac{68}{83}\right)\) |
| \(\chi_{167}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{70}{83}\right)\) | \(e\left(\frac{40}{83}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{45}{166}\right)\) | \(e\left(\frac{27}{83}\right)\) | \(e\left(\frac{82}{83}\right)\) | \(e\left(\frac{44}{83}\right)\) | \(e\left(\frac{80}{83}\right)\) | \(e\left(\frac{19}{166}\right)\) | \(e\left(\frac{49}{83}\right)\) |
| \(\chi_{167}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{83}\right)\) | \(e\left(\frac{33}{83}\right)\) | \(e\left(\frac{74}{83}\right)\) | \(e\left(\frac{89}{166}\right)\) | \(e\left(\frac{70}{83}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{66}{83}\right)\) | \(e\left(\frac{163}{166}\right)\) | \(e\left(\frac{1}{83}\right)\) |
| \(\chi_{167}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{16}{83}\right)\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{101}{166}\right)\) | \(e\left(\frac{44}{83}\right)\) | \(e\left(\frac{66}{83}\right)\) | \(e\left(\frac{1}{83}\right)\) | \(e\left(\frac{32}{83}\right)\) | \(e\left(\frac{157}{166}\right)\) | \(e\left(\frac{3}{83}\right)\) |