from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8016, base_ring=CyclotomicField(332))
M = H._module
chi = DirichletCharacter(H, M([0,249,0,206]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,8016))
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.x | 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: | $\Q(\zeta_{332})$ |
| Fixed field: | Number field defined by a degree 332 polynomial (not computed) |
First 31 of 164 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8016}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{123}{332}\right)\) | \(e\left(\frac{119}{166}\right)\) | \(e\left(\frac{41}{332}\right)\) | \(e\left(\frac{53}{332}\right)\) | \(e\left(\frac{147}{166}\right)\) | \(e\left(\frac{79}{332}\right)\) | \(e\left(\frac{77}{83}\right)\) | \(e\left(\frac{123}{166}\right)\) | \(e\left(\frac{107}{332}\right)\) | \(e\left(\frac{70}{83}\right)\) |
| \(\chi_{8016}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{205}{332}\right)\) | \(e\left(\frac{143}{166}\right)\) | \(e\left(\frac{179}{332}\right)\) | \(e\left(\frac{199}{332}\right)\) | \(e\left(\frac{79}{166}\right)\) | \(e\left(\frac{21}{332}\right)\) | \(e\left(\frac{73}{83}\right)\) | \(e\left(\frac{39}{166}\right)\) | \(e\left(\frac{289}{332}\right)\) | \(e\left(\frac{6}{83}\right)\) |
| \(\chi_{8016}(109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{332}\right)\) | \(e\left(\frac{93}{166}\right)\) | \(e\left(\frac{265}{332}\right)\) | \(e\left(\frac{213}{332}\right)\) | \(e\left(\frac{27}{166}\right)\) | \(e\left(\frac{211}{332}\right)\) | \(e\left(\frac{26}{83}\right)\) | \(e\left(\frac{131}{166}\right)\) | \(e\left(\frac{311}{332}\right)\) | \(e\left(\frac{1}{83}\right)\) |
| \(\chi_{8016}(253,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{171}{332}\right)\) | \(e\left(\frac{129}{166}\right)\) | \(e\left(\frac{57}{332}\right)\) | \(e\left(\frac{17}{332}\right)\) | \(e\left(\frac{91}{166}\right)\) | \(e\left(\frac{207}{332}\right)\) | \(e\left(\frac{20}{83}\right)\) | \(e\left(\frac{5}{166}\right)\) | \(e\left(\frac{3}{332}\right)\) | \(e\left(\frac{71}{83}\right)\) |
| \(\chi_{8016}(277,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{332}\right)\) | \(e\left(\frac{91}{166}\right)\) | \(e\left(\frac{295}{332}\right)\) | \(e\left(\frac{187}{332}\right)\) | \(e\left(\frac{5}{166}\right)\) | \(e\left(\frac{285}{332}\right)\) | \(e\left(\frac{54}{83}\right)\) | \(e\left(\frac{55}{166}\right)\) | \(e\left(\frac{33}{332}\right)\) | \(e\left(\frac{34}{83}\right)\) |
| \(\chi_{8016}(301,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{327}{332}\right)\) | \(e\left(\frac{37}{166}\right)\) | \(e\left(\frac{109}{332}\right)\) | \(e\left(\frac{149}{332}\right)\) | \(e\left(\frac{75}{166}\right)\) | \(e\left(\frac{291}{332}\right)\) | \(e\left(\frac{63}{83}\right)\) | \(e\left(\frac{161}{166}\right)\) | \(e\left(\frac{163}{332}\right)\) | \(e\left(\frac{12}{83}\right)\) |
| \(\chi_{8016}(325,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{293}{332}\right)\) | \(e\left(\frac{23}{166}\right)\) | \(e\left(\frac{319}{332}\right)\) | \(e\left(\frac{299}{332}\right)\) | \(e\left(\frac{87}{166}\right)\) | \(e\left(\frac{145}{332}\right)\) | \(e\left(\frac{10}{83}\right)\) | \(e\left(\frac{127}{166}\right)\) | \(e\left(\frac{209}{332}\right)\) | \(e\left(\frac{77}{83}\right)\) |
| \(\chi_{8016}(349,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{332}\right)\) | \(e\left(\frac{5}{166}\right)\) | \(e\left(\frac{257}{332}\right)\) | \(e\left(\frac{65}{332}\right)\) | \(e\left(\frac{55}{166}\right)\) | \(e\left(\frac{147}{332}\right)\) | \(e\left(\frac{13}{83}\right)\) | \(e\left(\frac{107}{166}\right)\) | \(e\left(\frac{31}{332}\right)\) | \(e\left(\frac{42}{83}\right)\) |
| \(\chi_{8016}(373,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{332}\right)\) | \(e\left(\frac{89}{166}\right)\) | \(e\left(\frac{159}{332}\right)\) | \(e\left(\frac{327}{332}\right)\) | \(e\left(\frac{149}{166}\right)\) | \(e\left(\frac{193}{332}\right)\) | \(e\left(\frac{82}{83}\right)\) | \(e\left(\frac{145}{166}\right)\) | \(e\left(\frac{253}{332}\right)\) | \(e\left(\frac{67}{83}\right)\) |
| \(\chi_{8016}(445,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{332}\right)\) | \(e\left(\frac{113}{166}\right)\) | \(e\left(\frac{297}{332}\right)\) | \(e\left(\frac{141}{332}\right)\) | \(e\left(\frac{81}{166}\right)\) | \(e\left(\frac{135}{332}\right)\) | \(e\left(\frac{78}{83}\right)\) | \(e\left(\frac{61}{166}\right)\) | \(e\left(\frac{103}{332}\right)\) | \(e\left(\frac{3}{83}\right)\) |
| \(\chi_{8016}(469,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{317}{332}\right)\) | \(e\left(\frac{111}{166}\right)\) | \(e\left(\frac{327}{332}\right)\) | \(e\left(\frac{115}{332}\right)\) | \(e\left(\frac{59}{166}\right)\) | \(e\left(\frac{209}{332}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{151}{166}\right)\) | \(e\left(\frac{157}{332}\right)\) | \(e\left(\frac{36}{83}\right)\) |
| \(\chi_{8016}(493,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{323}{332}\right)\) | \(e\left(\frac{133}{166}\right)\) | \(e\left(\frac{329}{332}\right)\) | \(e\left(\frac{69}{332}\right)\) | \(e\left(\frac{135}{166}\right)\) | \(e\left(\frac{59}{332}\right)\) | \(e\left(\frac{47}{83}\right)\) | \(e\left(\frac{157}{166}\right)\) | \(e\left(\frac{227}{332}\right)\) | \(e\left(\frac{5}{83}\right)\) |
| \(\chi_{8016}(541,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{159}{332}\right)\) | \(e\left(\frac{85}{166}\right)\) | \(e\left(\frac{53}{332}\right)\) | \(e\left(\frac{109}{332}\right)\) | \(e\left(\frac{105}{166}\right)\) | \(e\left(\frac{175}{332}\right)\) | \(e\left(\frac{55}{83}\right)\) | \(e\left(\frac{159}{166}\right)\) | \(e\left(\frac{195}{332}\right)\) | \(e\left(\frac{50}{83}\right)\) |
| \(\chi_{8016}(637,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{263}{332}\right)\) | \(e\left(\frac{79}{166}\right)\) | \(e\left(\frac{309}{332}\right)\) | \(e\left(\frac{197}{332}\right)\) | \(e\left(\frac{39}{166}\right)\) | \(e\left(\frac{231}{332}\right)\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{97}{166}\right)\) | \(e\left(\frac{191}{332}\right)\) | \(e\left(\frac{66}{83}\right)\) |
| \(\chi_{8016}(661,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{153}{332}\right)\) | \(e\left(\frac{63}{166}\right)\) | \(e\left(\frac{51}{332}\right)\) | \(e\left(\frac{155}{332}\right)\) | \(e\left(\frac{29}{166}\right)\) | \(e\left(\frac{325}{332}\right)\) | \(e\left(\frac{31}{83}\right)\) | \(e\left(\frac{153}{166}\right)\) | \(e\left(\frac{125}{332}\right)\) | \(e\left(\frac{81}{83}\right)\) |
| \(\chi_{8016}(685,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{332}\right)\) | \(e\left(\frac{29}{166}\right)\) | \(e\left(\frac{229}{332}\right)\) | \(e\left(\frac{45}{332}\right)\) | \(e\left(\frac{153}{166}\right)\) | \(e\left(\frac{255}{332}\right)\) | \(e\left(\frac{9}{83}\right)\) | \(e\left(\frac{23}{166}\right)\) | \(e\left(\frac{47}{332}\right)\) | \(e\left(\frac{61}{83}\right)\) |
| \(\chi_{8016}(709,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{277}{332}\right)\) | \(e\left(\frac{75}{166}\right)\) | \(e\left(\frac{203}{332}\right)\) | \(e\left(\frac{311}{332}\right)\) | \(e\left(\frac{161}{166}\right)\) | \(e\left(\frac{213}{332}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{111}{166}\right)\) | \(e\left(\frac{133}{332}\right)\) | \(e\left(\frac{49}{83}\right)\) |
| \(\chi_{8016}(781,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{332}\right)\) | \(e\left(\frac{65}{166}\right)\) | \(e\left(\frac{21}{332}\right)\) | \(e\left(\frac{181}{332}\right)\) | \(e\left(\frac{51}{166}\right)\) | \(e\left(\frac{251}{332}\right)\) | \(e\left(\frac{3}{83}\right)\) | \(e\left(\frac{63}{166}\right)\) | \(e\left(\frac{71}{332}\right)\) | \(e\left(\frac{48}{83}\right)\) |
| \(\chi_{8016}(829,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{332}\right)\) | \(e\left(\frac{125}{166}\right)\) | \(e\left(\frac{117}{332}\right)\) | \(e\left(\frac{297}{332}\right)\) | \(e\left(\frac{47}{166}\right)\) | \(e\left(\frac{23}{332}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{19}{166}\right)\) | \(e\left(\frac{111}{332}\right)\) | \(e\left(\frac{54}{83}\right)\) |
| \(\chi_{8016}(925,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{332}\right)\) | \(e\left(\frac{47}{166}\right)\) | \(e\left(\frac{125}{332}\right)\) | \(e\left(\frac{113}{332}\right)\) | \(e\left(\frac{19}{166}\right)\) | \(e\left(\frac{87}{332}\right)\) | \(e\left(\frac{6}{83}\right)\) | \(e\left(\frac{43}{166}\right)\) | \(e\left(\frac{59}{332}\right)\) | \(e\left(\frac{13}{83}\right)\) |
| \(\chi_{8016}(973,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{332}\right)\) | \(e\left(\frac{21}{166}\right)\) | \(e\left(\frac{17}{332}\right)\) | \(e\left(\frac{273}{332}\right)\) | \(e\left(\frac{65}{166}\right)\) | \(e\left(\frac{219}{332}\right)\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{51}{166}\right)\) | \(e\left(\frac{263}{332}\right)\) | \(e\left(\frac{27}{83}\right)\) |
| \(\chi_{8016}(1045,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{257}{332}\right)\) | \(e\left(\frac{57}{166}\right)\) | \(e\left(\frac{307}{332}\right)\) | \(e\left(\frac{243}{332}\right)\) | \(e\left(\frac{129}{166}\right)\) | \(e\left(\frac{49}{332}\right)\) | \(e\left(\frac{32}{83}\right)\) | \(e\left(\frac{91}{166}\right)\) | \(e\left(\frac{121}{332}\right)\) | \(e\left(\frac{14}{83}\right)\) |
| \(\chi_{8016}(1069,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{247}{332}\right)\) | \(e\left(\frac{131}{166}\right)\) | \(e\left(\frac{193}{332}\right)\) | \(e\left(\frac{209}{332}\right)\) | \(e\left(\frac{113}{166}\right)\) | \(e\left(\frac{299}{332}\right)\) | \(e\left(\frac{75}{83}\right)\) | \(e\left(\frac{81}{166}\right)\) | \(e\left(\frac{115}{332}\right)\) | \(e\left(\frac{38}{83}\right)\) |
| \(\chi_{8016}(1093,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{332}\right)\) | \(e\left(\frac{99}{166}\right)\) | \(e\left(\frac{175}{332}\right)\) | \(e\left(\frac{291}{332}\right)\) | \(e\left(\frac{93}{166}\right)\) | \(e\left(\frac{321}{332}\right)\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{27}{166}\right)\) | \(e\left(\frac{149}{332}\right)\) | \(e\left(\frac{68}{83}\right)\) |
| \(\chi_{8016}(1141,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{313}{332}\right)\) | \(e\left(\frac{41}{166}\right)\) | \(e\left(\frac{215}{332}\right)\) | \(e\left(\frac{35}{332}\right)\) | \(e\left(\frac{119}{166}\right)\) | \(e\left(\frac{309}{332}\right)\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{147}{166}\right)\) | \(e\left(\frac{221}{332}\right)\) | \(e\left(\frac{29}{83}\right)\) |
| \(\chi_{8016}(1165,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{243}{332}\right)\) | \(e\left(\frac{61}{166}\right)\) | \(e\left(\frac{81}{332}\right)\) | \(e\left(\frac{129}{332}\right)\) | \(e\left(\frac{7}{166}\right)\) | \(e\left(\frac{67}{332}\right)\) | \(e\left(\frac{59}{83}\right)\) | \(e\left(\frac{77}{166}\right)\) | \(e\left(\frac{179}{332}\right)\) | \(e\left(\frac{31}{83}\right)\) |
| \(\chi_{8016}(1189,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{245}{332}\right)\) | \(e\left(\frac{13}{166}\right)\) | \(e\left(\frac{303}{332}\right)\) | \(e\left(\frac{3}{332}\right)\) | \(e\left(\frac{143}{166}\right)\) | \(e\left(\frac{17}{332}\right)\) | \(e\left(\frac{67}{83}\right)\) | \(e\left(\frac{79}{166}\right)\) | \(e\left(\frac{313}{332}\right)\) | \(e\left(\frac{76}{83}\right)\) |
| \(\chi_{8016}(1237,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{332}\right)\) | \(e\left(\frac{7}{166}\right)\) | \(e\left(\frac{227}{332}\right)\) | \(e\left(\frac{91}{332}\right)\) | \(e\left(\frac{77}{166}\right)\) | \(e\left(\frac{73}{332}\right)\) | \(e\left(\frac{68}{83}\right)\) | \(e\left(\frac{17}{166}\right)\) | \(e\left(\frac{309}{332}\right)\) | \(e\left(\frac{9}{83}\right)\) |
| \(\chi_{8016}(1261,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{275}{332}\right)\) | \(e\left(\frac{123}{166}\right)\) | \(e\left(\frac{313}{332}\right)\) | \(e\left(\frac{105}{332}\right)\) | \(e\left(\frac{25}{166}\right)\) | \(e\left(\frac{263}{332}\right)\) | \(e\left(\frac{21}{83}\right)\) | \(e\left(\frac{109}{166}\right)\) | \(e\left(\frac{331}{332}\right)\) | \(e\left(\frac{4}{83}\right)\) |
| \(\chi_{8016}(1309,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{315}{332}\right)\) | \(e\left(\frac{159}{166}\right)\) | \(e\left(\frac{105}{332}\right)\) | \(e\left(\frac{241}{332}\right)\) | \(e\left(\frac{89}{166}\right)\) | \(e\left(\frac{259}{332}\right)\) | \(e\left(\frac{15}{83}\right)\) | \(e\left(\frac{149}{166}\right)\) | \(e\left(\frac{23}{332}\right)\) | \(e\left(\frac{74}{83}\right)\) |
| \(\chi_{8016}(1333,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{105}{332}\right)\) | \(e\left(\frac{53}{166}\right)\) | \(e\left(\frac{35}{332}\right)\) | \(e\left(\frac{191}{332}\right)\) | \(e\left(\frac{85}{166}\right)\) | \(e\left(\frac{197}{332}\right)\) | \(e\left(\frac{5}{83}\right)\) | \(e\left(\frac{105}{166}\right)\) | \(e\left(\frac{229}{332}\right)\) | \(e\left(\frac{80}{83}\right)\) |