from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8016, base_ring=CyclotomicField(332))
M = H._module
chi = DirichletCharacter(H, M([0,249,0,132]))
chi.galois_orbit()
[g,chi] = znchar(Mod(61,8016))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8016\) | |
| Conductor: | \(2672\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(332\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2672.w | 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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8016}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{332}\right)\) | \(e\left(\frac{69}{166}\right)\) | \(e\left(\frac{293}{332}\right)\) | \(e\left(\frac{67}{332}\right)\) | \(e\left(\frac{6}{83}\right)\) | \(e\left(\frac{103}{332}\right)\) | \(e\left(\frac{143}{166}\right)\) | \(e\left(\frac{49}{166}\right)\) | \(e\left(\frac{295}{332}\right)\) | \(e\left(\frac{65}{83}\right)\) |
| \(\chi_{8016}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{332}\right)\) | \(e\left(\frac{147}{166}\right)\) | \(e\left(\frac{119}{332}\right)\) | \(e\left(\frac{85}{332}\right)\) | \(e\left(\frac{20}{83}\right)\) | \(e\left(\frac{205}{332}\right)\) | \(e\left(\frac{117}{166}\right)\) | \(e\left(\frac{25}{166}\right)\) | \(e\left(\frac{181}{332}\right)\) | \(e\left(\frac{23}{83}\right)\) |
| \(\chi_{8016}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{332}\right)\) | \(e\left(\frac{101}{166}\right)\) | \(e\left(\frac{311}{332}\right)\) | \(e\left(\frac{317}{332}\right)\) | \(e\left(\frac{16}{83}\right)\) | \(e\left(\frac{81}{332}\right)\) | \(e\left(\frac{77}{166}\right)\) | \(e\left(\frac{103}{166}\right)\) | \(e\left(\frac{261}{332}\right)\) | \(e\left(\frac{35}{83}\right)\) |
| \(\chi_{8016}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{165}{332}\right)\) | \(e\left(\frac{107}{166}\right)\) | \(e\left(\frac{221}{332}\right)\) | \(e\left(\frac{63}{332}\right)\) | \(e\left(\frac{49}{83}\right)\) | \(e\left(\frac{191}{332}\right)\) | \(e\left(\frac{75}{166}\right)\) | \(e\left(\frac{165}{166}\right)\) | \(e\left(\frac{99}{332}\right)\) | \(e\left(\frac{19}{83}\right)\) |
| \(\chi_{8016}(181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{332}\right)\) | \(e\left(\frac{135}{166}\right)\) | \(e\left(\frac{299}{332}\right)\) | \(e\left(\frac{261}{332}\right)\) | \(e\left(\frac{37}{83}\right)\) | \(e\left(\frac{317}{332}\right)\) | \(e\left(\frac{121}{166}\right)\) | \(e\left(\frac{67}{166}\right)\) | \(e\left(\frac{173}{332}\right)\) | \(e\left(\frac{55}{83}\right)\) |
| \(\chi_{8016}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{332}\right)\) | \(e\left(\frac{27}{166}\right)\) | \(e\left(\frac{93}{332}\right)\) | \(e\left(\frac{19}{332}\right)\) | \(e\left(\frac{24}{83}\right)\) | \(e\left(\frac{163}{332}\right)\) | \(e\left(\frac{157}{166}\right)\) | \(e\left(\frac{113}{166}\right)\) | \(e\left(\frac{267}{332}\right)\) | \(e\left(\frac{11}{83}\right)\) |
| \(\chi_{8016}(229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{332}\right)\) | \(e\left(\frac{151}{166}\right)\) | \(e\left(\frac{59}{332}\right)\) | \(e\left(\frac{137}{332}\right)\) | \(e\left(\frac{42}{83}\right)\) | \(e\left(\frac{57}{332}\right)\) | \(e\left(\frac{5}{166}\right)\) | \(e\left(\frac{11}{166}\right)\) | \(e\left(\frac{73}{332}\right)\) | \(e\left(\frac{40}{83}\right)\) |
| \(\chi_{8016}(397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{197}{332}\right)\) | \(e\left(\frac{3}{166}\right)\) | \(e\left(\frac{121}{332}\right)\) | \(e\left(\frac{39}{332}\right)\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{55}{332}\right)\) | \(e\left(\frac{165}{166}\right)\) | \(e\left(\frac{31}{166}\right)\) | \(e\left(\frac{251}{332}\right)\) | \(e\left(\frac{75}{83}\right)\) |
| \(\chi_{8016}(421,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{239}{332}\right)\) | \(e\left(\frac{157}{166}\right)\) | \(e\left(\frac{135}{332}\right)\) | \(e\left(\frac{49}{332}\right)\) | \(e\left(\frac{75}{83}\right)\) | \(e\left(\frac{1}{332}\right)\) | \(e\left(\frac{3}{166}\right)\) | \(e\left(\frac{73}{166}\right)\) | \(e\left(\frac{77}{332}\right)\) | \(e\left(\frac{24}{83}\right)\) |
| \(\chi_{8016}(517,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{332}\right)\) | \(e\left(\frac{39}{166}\right)\) | \(e\left(\frac{79}{332}\right)\) | \(e\left(\frac{9}{332}\right)\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{217}{332}\right)\) | \(e\left(\frac{153}{166}\right)\) | \(e\left(\frac{71}{166}\right)\) | \(e\left(\frac{109}{332}\right)\) | \(e\left(\frac{62}{83}\right)\) |
| \(\chi_{8016}(565,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{231}{332}\right)\) | \(e\left(\frac{17}{166}\right)\) | \(e\left(\frac{243}{332}\right)\) | \(e\left(\frac{221}{332}\right)\) | \(e\left(\frac{52}{83}\right)\) | \(e\left(\frac{201}{332}\right)\) | \(e\left(\frac{105}{166}\right)\) | \(e\left(\frac{65}{166}\right)\) | \(e\left(\frac{205}{332}\right)\) | \(e\left(\frac{10}{83}\right)\) |
| \(\chi_{8016}(589,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{213}{332}\right)\) | \(e\left(\frac{117}{166}\right)\) | \(e\left(\frac{237}{332}\right)\) | \(e\left(\frac{27}{332}\right)\) | \(e\left(\frac{21}{83}\right)\) | \(e\left(\frac{319}{332}\right)\) | \(e\left(\frac{127}{166}\right)\) | \(e\left(\frac{47}{166}\right)\) | \(e\left(\frac{327}{332}\right)\) | \(e\left(\frac{20}{83}\right)\) |
| \(\chi_{8016}(613,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{307}{332}\right)\) | \(e\left(\frac{19}{166}\right)\) | \(e\left(\frac{47}{332}\right)\) | \(e\left(\frac{81}{332}\right)\) | \(e\left(\frac{63}{83}\right)\) | \(e\left(\frac{293}{332}\right)\) | \(e\left(\frac{49}{166}\right)\) | \(e\left(\frac{141}{166}\right)\) | \(e\left(\frac{317}{332}\right)\) | \(e\left(\frac{60}{83}\right)\) |
| \(\chi_{8016}(733,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{332}\right)\) | \(e\left(\frac{71}{166}\right)\) | \(e\left(\frac{97}{332}\right)\) | \(e\left(\frac{259}{332}\right)\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{195}{332}\right)\) | \(e\left(\frac{87}{166}\right)\) | \(e\left(\frac{125}{166}\right)\) | \(e\left(\frac{75}{332}\right)\) | \(e\left(\frac{32}{83}\right)\) |
| \(\chi_{8016}(757,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{332}\right)\) | \(e\left(\frac{161}{166}\right)\) | \(e\left(\frac{75}{332}\right)\) | \(e\left(\frac{101}{332}\right)\) | \(e\left(\frac{14}{83}\right)\) | \(e\left(\frac{185}{332}\right)\) | \(e\left(\frac{57}{166}\right)\) | \(e\left(\frac{59}{166}\right)\) | \(e\left(\frac{301}{332}\right)\) | \(e\left(\frac{41}{83}\right)\) |
| \(\chi_{8016}(805,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{187}{332}\right)\) | \(e\left(\frac{77}{166}\right)\) | \(e\left(\frac{7}{332}\right)\) | \(e\left(\frac{5}{332}\right)\) | \(e\left(\frac{50}{83}\right)\) | \(e\left(\frac{305}{332}\right)\) | \(e\left(\frac{85}{166}\right)\) | \(e\left(\frac{21}{166}\right)\) | \(e\left(\frac{245}{332}\right)\) | \(e\left(\frac{16}{83}\right)\) |
| \(\chi_{8016}(853,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{207}{332}\right)\) | \(e\left(\frac{95}{166}\right)\) | \(e\left(\frac{235}{332}\right)\) | \(e\left(\frac{73}{332}\right)\) | \(e\left(\frac{66}{83}\right)\) | \(e\left(\frac{137}{332}\right)\) | \(e\left(\frac{79}{166}\right)\) | \(e\left(\frac{41}{166}\right)\) | \(e\left(\frac{257}{332}\right)\) | \(e\left(\frac{51}{83}\right)\) |
| \(\chi_{8016}(877,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{332}\right)\) | \(e\left(\frac{105}{166}\right)\) | \(e\left(\frac{85}{332}\right)\) | \(e\left(\frac{203}{332}\right)\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{99}{332}\right)\) | \(e\left(\frac{131}{166}\right)\) | \(e\left(\frac{89}{166}\right)\) | \(e\left(\frac{319}{332}\right)\) | \(e\left(\frac{52}{83}\right)\) |
| \(\chi_{8016}(901,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{75}{332}\right)\) | \(e\left(\frac{109}{166}\right)\) | \(e\left(\frac{191}{332}\right)\) | \(e\left(\frac{89}{332}\right)\) | \(e\left(\frac{60}{83}\right)\) | \(e\left(\frac{117}{332}\right)\) | \(e\left(\frac{19}{166}\right)\) | \(e\left(\frac{75}{166}\right)\) | \(e\left(\frac{45}{332}\right)\) | \(e\left(\frac{69}{83}\right)\) |
| \(\chi_{8016}(949,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{135}{332}\right)\) | \(e\left(\frac{163}{166}\right)\) | \(e\left(\frac{211}{332}\right)\) | \(e\left(\frac{293}{332}\right)\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{277}{332}\right)\) | \(e\left(\frac{1}{166}\right)\) | \(e\left(\frac{135}{166}\right)\) | \(e\left(\frac{81}{332}\right)\) | \(e\left(\frac{8}{83}\right)\) |
| \(\chi_{8016}(997,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{332}\right)\) | \(e\left(\frac{35}{166}\right)\) | \(e\left(\frac{139}{332}\right)\) | \(e\left(\frac{289}{332}\right)\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{33}{332}\right)\) | \(e\left(\frac{99}{166}\right)\) | \(e\left(\frac{85}{166}\right)\) | \(e\left(\frac{217}{332}\right)\) | \(e\left(\frac{45}{83}\right)\) |
| \(\chi_{8016}(1021,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{332}\right)\) | \(e\left(\frac{121}{166}\right)\) | \(e\left(\frac{177}{332}\right)\) | \(e\left(\frac{79}{332}\right)\) | \(e\left(\frac{43}{83}\right)\) | \(e\left(\frac{171}{332}\right)\) | \(e\left(\frac{15}{166}\right)\) | \(e\left(\frac{33}{166}\right)\) | \(e\left(\frac{219}{332}\right)\) | \(e\left(\frac{37}{83}\right)\) |
| \(\chi_{8016}(1117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{117}{332}\right)\) | \(e\left(\frac{97}{166}\right)\) | \(e\left(\frac{205}{332}\right)\) | \(e\left(\frac{99}{332}\right)\) | \(e\left(\frac{77}{83}\right)\) | \(e\left(\frac{63}{332}\right)\) | \(e\left(\frac{23}{166}\right)\) | \(e\left(\frac{117}{166}\right)\) | \(e\left(\frac{203}{332}\right)\) | \(e\left(\frac{18}{83}\right)\) |
| \(\chi_{8016}(1213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{332}\right)\) | \(e\left(\frac{45}{166}\right)\) | \(e\left(\frac{321}{332}\right)\) | \(e\left(\frac{87}{332}\right)\) | \(e\left(\frac{40}{83}\right)\) | \(e\left(\frac{327}{332}\right)\) | \(e\left(\frac{151}{166}\right)\) | \(e\left(\frac{133}{166}\right)\) | \(e\left(\frac{279}{332}\right)\) | \(e\left(\frac{46}{83}\right)\) |
| \(\chi_{8016}(1285,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{211}{332}\right)\) | \(e\left(\frac{165}{166}\right)\) | \(e\left(\frac{15}{332}\right)\) | \(e\left(\frac{153}{332}\right)\) | \(e\left(\frac{36}{83}\right)\) | \(e\left(\frac{37}{332}\right)\) | \(e\left(\frac{111}{166}\right)\) | \(e\left(\frac{45}{166}\right)\) | \(e\left(\frac{193}{332}\right)\) | \(e\left(\frac{58}{83}\right)\) |
| \(\chi_{8016}(1357,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{332}\right)\) | \(e\left(\frac{33}{166}\right)\) | \(e\left(\frac{169}{332}\right)\) | \(e\left(\frac{263}{332}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{107}{332}\right)\) | \(e\left(\frac{155}{166}\right)\) | \(e\left(\frac{9}{166}\right)\) | \(e\left(\frac{271}{332}\right)\) | \(e\left(\frac{78}{83}\right)\) |
| \(\chi_{8016}(1429,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{332}\right)\) | \(e\left(\frac{49}{166}\right)\) | \(e\left(\frac{95}{332}\right)\) | \(e\left(\frac{305}{332}\right)\) | \(e\left(\frac{62}{83}\right)\) | \(e\left(\frac{13}{332}\right)\) | \(e\left(\frac{39}{166}\right)\) | \(e\left(\frac{119}{166}\right)\) | \(e\left(\frac{5}{332}\right)\) | \(e\left(\frac{63}{83}\right)\) |
| \(\chi_{8016}(1477,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{203}{332}\right)\) | \(e\left(\frac{25}{166}\right)\) | \(e\left(\frac{123}{332}\right)\) | \(e\left(\frac{325}{332}\right)\) | \(e\left(\frac{13}{83}\right)\) | \(e\left(\frac{237}{332}\right)\) | \(e\left(\frac{47}{166}\right)\) | \(e\left(\frac{37}{166}\right)\) | \(e\left(\frac{321}{332}\right)\) | \(e\left(\frac{44}{83}\right)\) |
| \(\chi_{8016}(1525,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{219}{332}\right)\) | \(e\left(\frac{139}{166}\right)\) | \(e\left(\frac{239}{332}\right)\) | \(e\left(\frac{313}{332}\right)\) | \(e\left(\frac{59}{83}\right)\) | \(e\left(\frac{169}{332}\right)\) | \(e\left(\frac{9}{166}\right)\) | \(e\left(\frac{53}{166}\right)\) | \(e\left(\frac{65}{332}\right)\) | \(e\left(\frac{72}{83}\right)\) |
| \(\chi_{8016}(1597,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{261}{332}\right)\) | \(e\left(\frac{127}{166}\right)\) | \(e\left(\frac{253}{332}\right)\) | \(e\left(\frac{323}{332}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{115}{332}\right)\) | \(e\left(\frac{13}{166}\right)\) | \(e\left(\frac{95}{166}\right)\) | \(e\left(\frac{223}{332}\right)\) | \(e\left(\frac{21}{83}\right)\) |
| \(\chi_{8016}(1717,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{332}\right)\) | \(e\left(\frac{55}{166}\right)\) | \(e\left(\frac{171}{332}\right)\) | \(e\left(\frac{217}{332}\right)\) | \(e\left(\frac{12}{83}\right)\) | \(e\left(\frac{289}{332}\right)\) | \(e\left(\frac{37}{166}\right)\) | \(e\left(\frac{15}{166}\right)\) | \(e\left(\frac{9}{332}\right)\) | \(e\left(\frac{47}{83}\right)\) |