from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8016, base_ring=CyclotomicField(166))
M = H._module
chi = DirichletCharacter(H, M([83,0,0,65]))
chi.galois_orbit()
[g,chi] = znchar(Mod(79,8016))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8016\) | |
| Conductor: | \(668\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(166\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 668.h | 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 166 polynomial (not computed) |
First 31 of 82 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8016}(79,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{166}\right)\) | \(e\left(\frac{117}{166}\right)\) | \(e\left(\frac{77}{166}\right)\) | \(e\left(\frac{55}{166}\right)\) | \(e\left(\frac{125}{166}\right)\) | \(e\left(\frac{35}{166}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{61}{83}\right)\) | \(e\left(\frac{123}{166}\right)\) |
| \(\chi_{8016}(271,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{166}\right)\) | \(e\left(\frac{3}{166}\right)\) | \(e\left(\frac{19}{166}\right)\) | \(e\left(\frac{61}{166}\right)\) | \(e\left(\frac{33}{166}\right)\) | \(e\left(\frac{69}{166}\right)\) | \(e\left(\frac{41}{83}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{42}{83}\right)\) | \(e\left(\frac{67}{166}\right)\) |
| \(\chi_{8016}(463,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{166}\right)\) | \(e\left(\frac{27}{166}\right)\) | \(e\left(\frac{5}{166}\right)\) | \(e\left(\frac{51}{166}\right)\) | \(e\left(\frac{131}{166}\right)\) | \(e\left(\frac{123}{166}\right)\) | \(e\left(\frac{37}{83}\right)\) | \(e\left(\frac{15}{83}\right)\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{105}{166}\right)\) |
| \(\chi_{8016}(511,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{166}\right)\) | \(e\left(\frac{107}{166}\right)\) | \(e\left(\frac{69}{166}\right)\) | \(e\left(\frac{73}{166}\right)\) | \(e\left(\frac{15}{166}\right)\) | \(e\left(\frac{137}{166}\right)\) | \(e\left(\frac{79}{83}\right)\) | \(e\left(\frac{41}{83}\right)\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{121}{166}\right)\) |
| \(\chi_{8016}(607,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{166}\right)\) | \(e\left(\frac{69}{166}\right)\) | \(e\left(\frac{105}{166}\right)\) | \(e\left(\frac{75}{166}\right)\) | \(e\left(\frac{95}{166}\right)\) | \(e\left(\frac{93}{166}\right)\) | \(e\left(\frac{30}{83}\right)\) | \(e\left(\frac{66}{83}\right)\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{47}{166}\right)\) |
| \(\chi_{8016}(703,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{166}\right)\) | \(e\left(\frac{15}{166}\right)\) | \(e\left(\frac{95}{166}\right)\) | \(e\left(\frac{139}{166}\right)\) | \(e\left(\frac{165}{166}\right)\) | \(e\left(\frac{13}{166}\right)\) | \(e\left(\frac{39}{83}\right)\) | \(e\left(\frac{36}{83}\right)\) | \(e\left(\frac{44}{83}\right)\) | \(e\left(\frac{3}{166}\right)\) |
| \(\chi_{8016}(751,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{166}\right)\) | \(e\left(\frac{11}{166}\right)\) | \(e\left(\frac{125}{166}\right)\) | \(e\left(\frac{113}{166}\right)\) | \(e\left(\frac{121}{166}\right)\) | \(e\left(\frac{87}{166}\right)\) | \(e\left(\frac{12}{83}\right)\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{71}{83}\right)\) | \(e\left(\frac{135}{166}\right)\) |
| \(\chi_{8016}(799,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{166}\right)\) | \(e\left(\frac{1}{166}\right)\) | \(e\left(\frac{117}{166}\right)\) | \(e\left(\frac{131}{166}\right)\) | \(e\left(\frac{11}{166}\right)\) | \(e\left(\frac{23}{166}\right)\) | \(e\left(\frac{69}{83}\right)\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{14}{83}\right)\) | \(e\left(\frac{133}{166}\right)\) |
| \(\chi_{8016}(895,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{166}\right)\) | \(e\left(\frac{149}{166}\right)\) | \(e\left(\frac{3}{166}\right)\) | \(e\left(\frac{97}{166}\right)\) | \(e\left(\frac{145}{166}\right)\) | \(e\left(\frac{107}{166}\right)\) | \(e\left(\frac{72}{83}\right)\) | \(e\left(\frac{9}{83}\right)\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{63}{166}\right)\) |
| \(\chi_{8016}(991,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{111}{166}\right)\) | \(e\left(\frac{67}{166}\right)\) | \(e\left(\frac{37}{166}\right)\) | \(e\left(\frac{145}{166}\right)\) | \(e\left(\frac{73}{166}\right)\) | \(e\left(\frac{47}{166}\right)\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{113}{166}\right)\) |
| \(\chi_{8016}(1039,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{166}\right)\) | \(e\left(\frac{143}{166}\right)\) | \(e\left(\frac{131}{166}\right)\) | \(e\left(\frac{141}{166}\right)\) | \(e\left(\frac{79}{166}\right)\) | \(e\left(\frac{135}{166}\right)\) | \(e\left(\frac{73}{83}\right)\) | \(e\left(\frac{61}{83}\right)\) | \(e\left(\frac{10}{83}\right)\) | \(e\left(\frac{95}{166}\right)\) |
| \(\chi_{8016}(1279,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{166}\right)\) | \(e\left(\frac{91}{166}\right)\) | \(e\left(\frac{23}{166}\right)\) | \(e\left(\frac{135}{166}\right)\) | \(e\left(\frac{5}{166}\right)\) | \(e\left(\frac{101}{166}\right)\) | \(e\left(\frac{54}{83}\right)\) | \(e\left(\frac{69}{83}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{151}{166}\right)\) |
| \(\chi_{8016}(1327,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{105}{166}\right)\) | \(e\left(\frac{23}{166}\right)\) | \(e\left(\frac{35}{166}\right)\) | \(e\left(\frac{25}{166}\right)\) | \(e\left(\frac{87}{166}\right)\) | \(e\left(\frac{31}{166}\right)\) | \(e\left(\frac{10}{83}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{73}{83}\right)\) | \(e\left(\frac{71}{166}\right)\) |
| \(\chi_{8016}(1375,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{166}\right)\) | \(e\left(\frac{89}{166}\right)\) | \(e\left(\frac{121}{166}\right)\) | \(e\left(\frac{39}{166}\right)\) | \(e\left(\frac{149}{166}\right)\) | \(e\left(\frac{55}{166}\right)\) | \(e\left(\frac{82}{83}\right)\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{1}{83}\right)\) | \(e\left(\frac{51}{166}\right)\) |
| \(\chi_{8016}(1471,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{117}{166}\right)\) | \(e\left(\frac{111}{166}\right)\) | \(e\left(\frac{39}{166}\right)\) | \(e\left(\frac{99}{166}\right)\) | \(e\left(\frac{59}{166}\right)\) | \(e\left(\frac{63}{166}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{34}{83}\right)\) | \(e\left(\frac{60}{83}\right)\) | \(e\left(\frac{155}{166}\right)\) |
| \(\chi_{8016}(1663,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{166}\right)\) | \(e\left(\frac{63}{166}\right)\) | \(e\left(\frac{67}{166}\right)\) | \(e\left(\frac{119}{166}\right)\) | \(e\left(\frac{29}{166}\right)\) | \(e\left(\frac{121}{166}\right)\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{35}{83}\right)\) | \(e\left(\frac{52}{83}\right)\) | \(e\left(\frac{79}{166}\right)\) |
| \(\chi_{8016}(1711,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{166}\right)\) | \(e\left(\frac{75}{166}\right)\) | \(e\left(\frac{143}{166}\right)\) | \(e\left(\frac{31}{166}\right)\) | \(e\left(\frac{161}{166}\right)\) | \(e\left(\frac{65}{166}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{14}{83}\right)\) | \(e\left(\frac{54}{83}\right)\) | \(e\left(\frac{15}{166}\right)\) |
| \(\chi_{8016}(2047,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{87}{166}\right)\) | \(e\left(\frac{57}{166}\right)\) | \(e\left(\frac{29}{166}\right)\) | \(e\left(\frac{163}{166}\right)\) | \(e\left(\frac{129}{166}\right)\) | \(e\left(\frac{149}{166}\right)\) | \(e\left(\frac{32}{83}\right)\) | \(e\left(\frac{4}{83}\right)\) | \(e\left(\frac{51}{83}\right)\) | \(e\left(\frac{111}{166}\right)\) |
| \(\chi_{8016}(2095,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{166}\right)\) | \(e\left(\frac{99}{166}\right)\) | \(e\left(\frac{129}{166}\right)\) | \(e\left(\frac{21}{166}\right)\) | \(e\left(\frac{93}{166}\right)\) | \(e\left(\frac{119}{166}\right)\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{55}{83}\right)\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{53}{166}\right)\) |
| \(\chi_{8016}(2143,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{166}\right)\) | \(e\left(\frac{41}{166}\right)\) | \(e\left(\frac{149}{166}\right)\) | \(e\left(\frac{59}{166}\right)\) | \(e\left(\frac{119}{166}\right)\) | \(e\left(\frac{113}{166}\right)\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{32}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{141}{166}\right)\) |
| \(\chi_{8016}(2191,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{166}\right)\) | \(e\left(\frac{13}{166}\right)\) | \(e\left(\frac{27}{166}\right)\) | \(e\left(\frac{43}{166}\right)\) | \(e\left(\frac{143}{166}\right)\) | \(e\left(\frac{133}{166}\right)\) | \(e\left(\frac{67}{83}\right)\) | \(e\left(\frac{81}{83}\right)\) | \(e\left(\frac{16}{83}\right)\) | \(e\left(\frac{69}{166}\right)\) |
| \(\chi_{8016}(2239,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{166}\right)\) | \(e\left(\frac{7}{166}\right)\) | \(e\left(\frac{155}{166}\right)\) | \(e\left(\frac{87}{166}\right)\) | \(e\left(\frac{77}{166}\right)\) | \(e\left(\frac{161}{166}\right)\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{50}{83}\right)\) | \(e\left(\frac{15}{83}\right)\) | \(e\left(\frac{101}{166}\right)\) |
| \(\chi_{8016}(2335,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{166}\right)\) | \(e\left(\frac{53}{166}\right)\) | \(e\left(\frac{59}{166}\right)\) | \(e\left(\frac{137}{166}\right)\) | \(e\left(\frac{85}{166}\right)\) | \(e\left(\frac{57}{166}\right)\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{78}{83}\right)\) | \(e\left(\frac{77}{166}\right)\) |
| \(\chi_{8016}(2383,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{166}\right)\) | \(e\left(\frac{141}{166}\right)\) | \(e\left(\frac{63}{166}\right)\) | \(e\left(\frac{45}{166}\right)\) | \(e\left(\frac{57}{166}\right)\) | \(e\left(\frac{89}{166}\right)\) | \(e\left(\frac{18}{83}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{161}{166}\right)\) |
| \(\chi_{8016}(2575,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{159}{166}\right)\) | \(e\left(\frac{87}{166}\right)\) | \(e\left(\frac{53}{166}\right)\) | \(e\left(\frac{109}{166}\right)\) | \(e\left(\frac{127}{166}\right)\) | \(e\left(\frac{9}{166}\right)\) | \(e\left(\frac{27}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{117}{166}\right)\) |
| \(\chi_{8016}(2623,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{153}{166}\right)\) | \(e\left(\frac{43}{166}\right)\) | \(e\left(\frac{51}{166}\right)\) | \(e\left(\frac{155}{166}\right)\) | \(e\left(\frac{141}{166}\right)\) | \(e\left(\frac{159}{166}\right)\) | \(e\left(\frac{62}{83}\right)\) | \(e\left(\frac{70}{83}\right)\) | \(e\left(\frac{21}{83}\right)\) | \(e\left(\frac{75}{166}\right)\) |
| \(\chi_{8016}(2767,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{166}\right)\) | \(e\left(\frac{73}{166}\right)\) | \(e\left(\frac{75}{166}\right)\) | \(e\left(\frac{101}{166}\right)\) | \(e\left(\frac{139}{166}\right)\) | \(e\left(\frac{19}{166}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{59}{83}\right)\) | \(e\left(\frac{26}{83}\right)\) | \(e\left(\frac{81}{166}\right)\) |
| \(\chi_{8016}(2815,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{166}\right)\) | \(e\left(\frac{103}{166}\right)\) | \(e\left(\frac{99}{166}\right)\) | \(e\left(\frac{47}{166}\right)\) | \(e\left(\frac{137}{166}\right)\) | \(e\left(\frac{45}{166}\right)\) | \(e\left(\frac{52}{83}\right)\) | \(e\left(\frac{48}{83}\right)\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{87}{166}\right)\) |
| \(\chi_{8016}(2959,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{166}\right)\) | \(e\left(\frac{55}{166}\right)\) | \(e\left(\frac{127}{166}\right)\) | \(e\left(\frac{67}{166}\right)\) | \(e\left(\frac{107}{166}\right)\) | \(e\left(\frac{103}{166}\right)\) | \(e\left(\frac{60}{83}\right)\) | \(e\left(\frac{49}{83}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{11}{166}\right)\) |
| \(\chi_{8016}(3151,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{166}\right)\) | \(e\left(\frac{139}{166}\right)\) | \(e\left(\frac{161}{166}\right)\) | \(e\left(\frac{115}{166}\right)\) | \(e\left(\frac{35}{166}\right)\) | \(e\left(\frac{43}{166}\right)\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{37}{83}\right)\) | \(e\left(\frac{61}{166}\right)\) |
| \(\chi_{8016}(3199,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{166}\right)\) | \(e\left(\frac{25}{166}\right)\) | \(e\left(\frac{103}{166}\right)\) | \(e\left(\frac{121}{166}\right)\) | \(e\left(\frac{109}{166}\right)\) | \(e\left(\frac{77}{166}\right)\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{60}{83}\right)\) | \(e\left(\frac{18}{83}\right)\) | \(e\left(\frac{5}{166}\right)\) |