from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8016, base_ring=CyclotomicField(332))
M = H._module
chi = DirichletCharacter(H, M([0,83,166,2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,8016))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8016\) | |
| Conductor: | \(8016\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(332\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{332})$ |
| Fixed field: | Number field defined by a degree 332 polynomial (not computed) |
First 31 of 164 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8016}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{332}\right)\) | \(e\left(\frac{35}{166}\right)\) | \(e\left(\frac{305}{332}\right)\) | \(e\left(\frac{123}{332}\right)\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{33}{332}\right)\) | \(e\left(\frac{99}{166}\right)\) | \(e\left(\frac{85}{166}\right)\) | \(e\left(\frac{51}{332}\right)\) | \(e\left(\frac{45}{83}\right)\) |
| \(\chi_{8016}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{135}{332}\right)\) | \(e\left(\frac{163}{166}\right)\) | \(e\left(\frac{45}{332}\right)\) | \(e\left(\frac{127}{332}\right)\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{277}{332}\right)\) | \(e\left(\frac{1}{166}\right)\) | \(e\left(\frac{135}{166}\right)\) | \(e\left(\frac{247}{332}\right)\) | \(e\left(\frac{8}{83}\right)\) |
| \(\chi_{8016}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{75}{332}\right)\) | \(e\left(\frac{109}{166}\right)\) | \(e\left(\frac{25}{332}\right)\) | \(e\left(\frac{255}{332}\right)\) | \(e\left(\frac{60}{83}\right)\) | \(e\left(\frac{117}{332}\right)\) | \(e\left(\frac{19}{166}\right)\) | \(e\left(\frac{75}{166}\right)\) | \(e\left(\frac{211}{332}\right)\) | \(e\left(\frac{69}{83}\right)\) |
| \(\chi_{8016}(125,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{332}\right)\) | \(e\left(\frac{105}{166}\right)\) | \(e\left(\frac{251}{332}\right)\) | \(e\left(\frac{37}{332}\right)\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{99}{332}\right)\) | \(e\left(\frac{131}{166}\right)\) | \(e\left(\frac{89}{166}\right)\) | \(e\left(\frac{153}{332}\right)\) | \(e\left(\frac{52}{83}\right)\) |
| \(\chi_{8016}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{207}{332}\right)\) | \(e\left(\frac{95}{166}\right)\) | \(e\left(\frac{69}{332}\right)\) | \(e\left(\frac{239}{332}\right)\) | \(e\left(\frac{66}{83}\right)\) | \(e\left(\frac{137}{332}\right)\) | \(e\left(\frac{79}{166}\right)\) | \(e\left(\frac{41}{166}\right)\) | \(e\left(\frac{91}{332}\right)\) | \(e\left(\frac{51}{83}\right)\) |
| \(\chi_{8016}(197,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{187}{332}\right)\) | \(e\left(\frac{77}{166}\right)\) | \(e\left(\frac{173}{332}\right)\) | \(e\left(\frac{171}{332}\right)\) | \(e\left(\frac{50}{83}\right)\) | \(e\left(\frac{305}{332}\right)\) | \(e\left(\frac{85}{166}\right)\) | \(e\left(\frac{21}{166}\right)\) | \(e\left(\frac{79}{332}\right)\) | \(e\left(\frac{16}{83}\right)\) |
| \(\chi_{8016}(245,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{332}\right)\) | \(e\left(\frac{161}{166}\right)\) | \(e\left(\frac{241}{332}\right)\) | \(e\left(\frac{267}{332}\right)\) | \(e\left(\frac{14}{83}\right)\) | \(e\left(\frac{185}{332}\right)\) | \(e\left(\frac{57}{166}\right)\) | \(e\left(\frac{59}{166}\right)\) | \(e\left(\frac{135}{332}\right)\) | \(e\left(\frac{41}{83}\right)\) |
| \(\chi_{8016}(269,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{332}\right)\) | \(e\left(\frac{71}{166}\right)\) | \(e\left(\frac{263}{332}\right)\) | \(e\left(\frac{93}{332}\right)\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{195}{332}\right)\) | \(e\left(\frac{87}{166}\right)\) | \(e\left(\frac{125}{166}\right)\) | \(e\left(\frac{241}{332}\right)\) | \(e\left(\frac{32}{83}\right)\) |
| \(\chi_{8016}(389,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{307}{332}\right)\) | \(e\left(\frac{19}{166}\right)\) | \(e\left(\frac{213}{332}\right)\) | \(e\left(\frac{247}{332}\right)\) | \(e\left(\frac{63}{83}\right)\) | \(e\left(\frac{293}{332}\right)\) | \(e\left(\frac{49}{166}\right)\) | \(e\left(\frac{141}{166}\right)\) | \(e\left(\frac{151}{332}\right)\) | \(e\left(\frac{60}{83}\right)\) |
| \(\chi_{8016}(413,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{213}{332}\right)\) | \(e\left(\frac{117}{166}\right)\) | \(e\left(\frac{71}{332}\right)\) | \(e\left(\frac{193}{332}\right)\) | \(e\left(\frac{21}{83}\right)\) | \(e\left(\frac{319}{332}\right)\) | \(e\left(\frac{127}{166}\right)\) | \(e\left(\frac{47}{166}\right)\) | \(e\left(\frac{161}{332}\right)\) | \(e\left(\frac{20}{83}\right)\) |
| \(\chi_{8016}(437,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{231}{332}\right)\) | \(e\left(\frac{17}{166}\right)\) | \(e\left(\frac{77}{332}\right)\) | \(e\left(\frac{55}{332}\right)\) | \(e\left(\frac{52}{83}\right)\) | \(e\left(\frac{201}{332}\right)\) | \(e\left(\frac{105}{166}\right)\) | \(e\left(\frac{65}{166}\right)\) | \(e\left(\frac{39}{332}\right)\) | \(e\left(\frac{10}{83}\right)\) |
| \(\chi_{8016}(485,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{332}\right)\) | \(e\left(\frac{39}{166}\right)\) | \(e\left(\frac{245}{332}\right)\) | \(e\left(\frac{175}{332}\right)\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{217}{332}\right)\) | \(e\left(\frac{153}{166}\right)\) | \(e\left(\frac{71}{166}\right)\) | \(e\left(\frac{275}{332}\right)\) | \(e\left(\frac{62}{83}\right)\) |
| \(\chi_{8016}(581,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{239}{332}\right)\) | \(e\left(\frac{157}{166}\right)\) | \(e\left(\frac{301}{332}\right)\) | \(e\left(\frac{215}{332}\right)\) | \(e\left(\frac{75}{83}\right)\) | \(e\left(\frac{1}{332}\right)\) | \(e\left(\frac{3}{166}\right)\) | \(e\left(\frac{73}{166}\right)\) | \(e\left(\frac{243}{332}\right)\) | \(e\left(\frac{24}{83}\right)\) |
| \(\chi_{8016}(605,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{197}{332}\right)\) | \(e\left(\frac{3}{166}\right)\) | \(e\left(\frac{287}{332}\right)\) | \(e\left(\frac{205}{332}\right)\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{55}{332}\right)\) | \(e\left(\frac{165}{166}\right)\) | \(e\left(\frac{31}{166}\right)\) | \(e\left(\frac{85}{332}\right)\) | \(e\left(\frac{75}{83}\right)\) |
| \(\chi_{8016}(773,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{332}\right)\) | \(e\left(\frac{151}{166}\right)\) | \(e\left(\frac{225}{332}\right)\) | \(e\left(\frac{303}{332}\right)\) | \(e\left(\frac{42}{83}\right)\) | \(e\left(\frac{57}{332}\right)\) | \(e\left(\frac{5}{166}\right)\) | \(e\left(\frac{11}{166}\right)\) | \(e\left(\frac{239}{332}\right)\) | \(e\left(\frac{40}{83}\right)\) |
| \(\chi_{8016}(797,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{332}\right)\) | \(e\left(\frac{27}{166}\right)\) | \(e\left(\frac{259}{332}\right)\) | \(e\left(\frac{185}{332}\right)\) | \(e\left(\frac{24}{83}\right)\) | \(e\left(\frac{163}{332}\right)\) | \(e\left(\frac{157}{166}\right)\) | \(e\left(\frac{113}{166}\right)\) | \(e\left(\frac{101}{332}\right)\) | \(e\left(\frac{11}{83}\right)\) |
| \(\chi_{8016}(821,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{332}\right)\) | \(e\left(\frac{135}{166}\right)\) | \(e\left(\frac{133}{332}\right)\) | \(e\left(\frac{95}{332}\right)\) | \(e\left(\frac{37}{83}\right)\) | \(e\left(\frac{317}{332}\right)\) | \(e\left(\frac{121}{166}\right)\) | \(e\left(\frac{67}{166}\right)\) | \(e\left(\frac{7}{332}\right)\) | \(e\left(\frac{55}{83}\right)\) |
| \(\chi_{8016}(845,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{165}{332}\right)\) | \(e\left(\frac{107}{166}\right)\) | \(e\left(\frac{55}{332}\right)\) | \(e\left(\frac{229}{332}\right)\) | \(e\left(\frac{49}{83}\right)\) | \(e\left(\frac{191}{332}\right)\) | \(e\left(\frac{75}{166}\right)\) | \(e\left(\frac{165}{166}\right)\) | \(e\left(\frac{265}{332}\right)\) | \(e\left(\frac{19}{83}\right)\) |
| \(\chi_{8016}(869,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{332}\right)\) | \(e\left(\frac{101}{166}\right)\) | \(e\left(\frac{145}{332}\right)\) | \(e\left(\frac{151}{332}\right)\) | \(e\left(\frac{16}{83}\right)\) | \(e\left(\frac{81}{332}\right)\) | \(e\left(\frac{77}{166}\right)\) | \(e\left(\frac{103}{166}\right)\) | \(e\left(\frac{95}{332}\right)\) | \(e\left(\frac{35}{83}\right)\) |
| \(\chi_{8016}(917,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{332}\right)\) | \(e\left(\frac{147}{166}\right)\) | \(e\left(\frac{285}{332}\right)\) | \(e\left(\frac{251}{332}\right)\) | \(e\left(\frac{20}{83}\right)\) | \(e\left(\frac{205}{332}\right)\) | \(e\left(\frac{117}{166}\right)\) | \(e\left(\frac{25}{166}\right)\) | \(e\left(\frac{15}{332}\right)\) | \(e\left(\frac{23}{83}\right)\) |
| \(\chi_{8016}(941,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{332}\right)\) | \(e\left(\frac{69}{166}\right)\) | \(e\left(\frac{127}{332}\right)\) | \(e\left(\frac{233}{332}\right)\) | \(e\left(\frac{6}{83}\right)\) | \(e\left(\frac{103}{332}\right)\) | \(e\left(\frac{143}{166}\right)\) | \(e\left(\frac{49}{166}\right)\) | \(e\left(\frac{129}{332}\right)\) | \(e\left(\frac{65}{83}\right)\) |
| \(\chi_{8016}(1037,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{321}{332}\right)\) | \(e\left(\frac{15}{166}\right)\) | \(e\left(\frac{107}{332}\right)\) | \(e\left(\frac{29}{332}\right)\) | \(e\left(\frac{41}{83}\right)\) | \(e\left(\frac{275}{332}\right)\) | \(e\left(\frac{161}{166}\right)\) | \(e\left(\frac{155}{166}\right)\) | \(e\left(\frac{93}{332}\right)\) | \(e\left(\frac{43}{83}\right)\) |
| \(\chi_{8016}(1061,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{332}\right)\) | \(e\left(\frac{137}{166}\right)\) | \(e\left(\frac{269}{332}\right)\) | \(e\left(\frac{287}{332}\right)\) | \(e\left(\frac{48}{83}\right)\) | \(e\left(\frac{77}{332}\right)\) | \(e\left(\frac{65}{166}\right)\) | \(e\left(\frac{143}{166}\right)\) | \(e\left(\frac{119}{332}\right)\) | \(e\left(\frac{22}{83}\right)\) |
| \(\chi_{8016}(1085,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{169}{332}\right)\) | \(e\left(\frac{11}{166}\right)\) | \(e\left(\frac{167}{332}\right)\) | \(e\left(\frac{309}{332}\right)\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{91}{332}\right)\) | \(e\left(\frac{107}{166}\right)\) | \(e\left(\frac{3}{166}\right)\) | \(e\left(\frac{201}{332}\right)\) | \(e\left(\frac{26}{83}\right)\) |
| \(\chi_{8016}(1133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{332}\right)\) | \(e\left(\frac{1}{166}\right)\) | \(e\left(\frac{151}{332}\right)\) | \(e\left(\frac{13}{332}\right)\) | \(e\left(\frac{47}{83}\right)\) | \(e\left(\frac{295}{332}\right)\) | \(e\left(\frac{55}{166}\right)\) | \(e\left(\frac{121}{166}\right)\) | \(e\left(\frac{305}{332}\right)\) | \(e\left(\frac{25}{83}\right)\) |
| \(\chi_{8016}(1157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{99}{332}\right)\) | \(e\left(\frac{31}{166}\right)\) | \(e\left(\frac{33}{332}\right)\) | \(e\left(\frac{71}{332}\right)\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{181}{332}\right)\) | \(e\left(\frac{45}{166}\right)\) | \(e\left(\frac{99}{166}\right)\) | \(e\left(\frac{159}{332}\right)\) | \(e\left(\frac{28}{83}\right)\) |
| \(\chi_{8016}(1229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{332}\right)\) | \(e\left(\frac{149}{166}\right)\) | \(e\left(\frac{255}{332}\right)\) | \(e\left(\frac{277}{332}\right)\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{131}{332}\right)\) | \(e\left(\frac{61}{166}\right)\) | \(e\left(\frac{101}{166}\right)\) | \(e\left(\frac{293}{332}\right)\) | \(e\left(\frac{73}{83}\right)\) |
| \(\chi_{8016}(1325,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{305}{332}\right)\) | \(e\left(\frac{67}{166}\right)\) | \(e\left(\frac{323}{332}\right)\) | \(e\left(\frac{41}{332}\right)\) | \(e\left(\frac{78}{83}\right)\) | \(e\left(\frac{11}{332}\right)\) | \(e\left(\frac{33}{166}\right)\) | \(e\left(\frac{139}{166}\right)\) | \(e\left(\frac{17}{332}\right)\) | \(e\left(\frac{15}{83}\right)\) |
| \(\chi_{8016}(1349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{123}{332}\right)\) | \(e\left(\frac{119}{166}\right)\) | \(e\left(\frac{41}{332}\right)\) | \(e\left(\frac{219}{332}\right)\) | \(e\left(\frac{32}{83}\right)\) | \(e\left(\frac{245}{332}\right)\) | \(e\left(\frac{71}{166}\right)\) | \(e\left(\frac{123}{166}\right)\) | \(e\left(\frac{107}{332}\right)\) | \(e\left(\frac{70}{83}\right)\) |
| \(\chi_{8016}(1373,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{205}{332}\right)\) | \(e\left(\frac{143}{166}\right)\) | \(e\left(\frac{179}{332}\right)\) | \(e\left(\frac{33}{332}\right)\) | \(e\left(\frac{81}{83}\right)\) | \(e\left(\frac{187}{332}\right)\) | \(e\left(\frac{63}{166}\right)\) | \(e\left(\frac{39}{166}\right)\) | \(e\left(\frac{289}{332}\right)\) | \(e\left(\frac{6}{83}\right)\) |
| \(\chi_{8016}(1445,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{332}\right)\) | \(e\left(\frac{93}{166}\right)\) | \(e\left(\frac{265}{332}\right)\) | \(e\left(\frac{47}{332}\right)\) | \(e\left(\frac{55}{83}\right)\) | \(e\left(\frac{45}{332}\right)\) | \(e\left(\frac{135}{166}\right)\) | \(e\left(\frac{131}{166}\right)\) | \(e\left(\frac{311}{332}\right)\) | \(e\left(\frac{1}{83}\right)\) |