from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3963, base_ring=CyclotomicField(1320))
M = H._module
chi = DirichletCharacter(H, M([0,1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,3963))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3963\) | |
| Conductor: | \(1321\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1320\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1321.bf | 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: | $\Q(\zeta_{1320})$ |
| Fixed field: | Number field defined by a degree 1320 polynomial (not computed) |
First 31 of 320 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3963}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{551}{660}\right)\) | \(e\left(\frac{113}{264}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{1}{165}\right)\) | \(e\left(\frac{1}{1320}\right)\) | \(e\left(\frac{269}{440}\right)\) | \(e\left(\frac{11}{15}\right)\) |
| \(\chi_{3963}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{469}{660}\right)\) | \(e\left(\frac{43}{264}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{29}{165}\right)\) | \(e\left(\frac{1019}{1320}\right)\) | \(e\left(\frac{431}{440}\right)\) | \(e\left(\frac{4}{15}\right)\) |
| \(\chi_{3963}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{499}{660}\right)\) | \(e\left(\frac{1}{264}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{1049}{1320}\right)\) | \(e\left(\frac{141}{440}\right)\) | \(e\left(\frac{4}{15}\right)\) |
| \(\chi_{3963}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{479}{660}\right)\) | \(e\left(\frac{29}{264}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{589}{1320}\right)\) | \(e\left(\frac{41}{440}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{3963}(85,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{607}{660}\right)\) | \(e\left(\frac{193}{264}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{2}{165}\right)\) | \(e\left(\frac{497}{1320}\right)\) | \(e\left(\frac{373}{440}\right)\) | \(e\left(\frac{7}{15}\right)\) |
| \(\chi_{3963}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{299}{660}\right)\) | \(e\left(\frac{149}{264}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{1069}{1320}\right)\) | \(e\left(\frac{241}{440}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{3963}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{419}{660}\right)\) | \(e\left(\frac{113}{264}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{34}{165}\right)\) | \(e\left(\frac{529}{1320}\right)\) | \(e\left(\frac{181}{440}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{3963}(115,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{73}{660}\right)\) | \(e\left(\frac{175}{264}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{623}{1320}\right)\) | \(e\left(\frac{387}{440}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{3963}(118,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{283}{660}\right)\) | \(e\left(\frac{13}{264}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{8}{165}\right)\) | \(e\left(\frac{173}{1320}\right)\) | \(e\left(\frac{337}{440}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{3963}(139,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{191}{660}\right)\) | \(e\left(\frac{89}{264}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{961}{1320}\right)\) | \(e\left(\frac{229}{440}\right)\) | \(e\left(\frac{11}{15}\right)\) |
| \(\chi_{3963}(151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{203}{660}\right)\) | \(e\left(\frac{125}{264}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{973}{1320}\right)\) | \(e\left(\frac{377}{440}\right)\) | \(e\left(\frac{8}{15}\right)\) |
| \(\chi_{3963}(157,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{487}{660}\right)\) | \(e\left(\frac{229}{264}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{47}{165}\right)\) | \(e\left(\frac{1037}{1320}\right)\) | \(e\left(\frac{433}{440}\right)\) | \(e\left(\frac{7}{15}\right)\) |
| \(\chi_{3963}(166,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{389}{660}\right)\) | \(e\left(\frac{23}{264}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{4}{165}\right)\) | \(e\left(\frac{1159}{1320}\right)\) | \(e\left(\frac{251}{440}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{3963}(178,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{353}{660}\right)\) | \(e\left(\frac{179}{264}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{133}{165}\right)\) | \(e\left(\frac{1123}{1320}\right)\) | \(e\left(\frac{247}{440}\right)\) | \(e\left(\frac{8}{15}\right)\) |
| \(\chi_{3963}(181,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{659}{660}\right)\) | \(e\left(\frac{41}{264}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{769}{1320}\right)\) | \(e\left(\frac{61}{440}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{3963}(184,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{137}{660}\right)\) | \(e\left(\frac{191}{264}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{247}{1320}\right)\) | \(e\left(\frac{3}{440}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{3963}(190,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{493}{660}\right)\) | \(e\left(\frac{115}{264}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{53}{165}\right)\) | \(e\left(\frac{1043}{1320}\right)\) | \(e\left(\frac{287}{440}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{3963}(193,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{373}{660}\right)\) | \(e\left(\frac{19}{264}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{923}{1320}\right)\) | \(e\left(\frac{127}{440}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{3963}(199,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{559}{660}\right)\) | \(e\left(\frac{49}{264}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{119}{165}\right)\) | \(e\left(\frac{449}{1320}\right)\) | \(e\left(\frac{221}{440}\right)\) | \(e\left(\frac{4}{15}\right)\) |
| \(\chi_{3963}(202,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{197}{660}\right)\) | \(e\left(\frac{107}{264}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{142}{165}\right)\) | \(e\left(\frac{307}{1320}\right)\) | \(e\left(\frac{303}{440}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{3963}(223,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{439}{660}\right)\) | \(e\left(\frac{217}{264}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{329}{1320}\right)\) | \(e\left(\frac{61}{440}\right)\) | \(e\left(\frac{4}{15}\right)\) |
| \(\chi_{3963}(244,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{173}{660}\right)\) | \(e\left(\frac{167}{264}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{118}{165}\right)\) | \(e\left(\frac{943}{1320}\right)\) | \(e\left(\frac{227}{440}\right)\) | \(e\left(\frac{8}{15}\right)\) |
| \(\chi_{3963}(259,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{571}{660}\right)\) | \(e\left(\frac{217}{264}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{1121}{1320}\right)\) | \(e\left(\frac{149}{440}\right)\) | \(e\left(\frac{1}{15}\right)\) |
| \(\chi_{3963}(268,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{467}{660}\right)\) | \(e\left(\frac{125}{264}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{1237}{1320}\right)\) | \(e\left(\frac{113}{440}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{3963}(280,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{523}{660}\right)\) | \(e\left(\frac{73}{264}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{83}{165}\right)\) | \(e\left(\frac{1073}{1320}\right)\) | \(e\left(\frac{437}{440}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{3963}(286,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{361}{660}\right)\) | \(e\left(\frac{115}{264}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{86}{165}\right)\) | \(e\left(\frac{251}{1320}\right)\) | \(e\left(\frac{199}{440}\right)\) | \(e\left(\frac{1}{15}\right)\) |
| \(\chi_{3963}(295,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{263}{660}\right)\) | \(e\left(\frac{41}{264}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{43}{165}\right)\) | \(e\left(\frac{1033}{1320}\right)\) | \(e\left(\frac{237}{440}\right)\) | \(e\left(\frac{8}{15}\right)\) |
| \(\chi_{3963}(304,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{557}{660}\right)\) | \(e\left(\frac{131}{264}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{667}{1320}\right)\) | \(e\left(\frac{343}{440}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{3963}(316,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{569}{660}\right)\) | \(e\left(\frac{35}{264}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{19}{1320}\right)\) | \(e\left(\frac{271}{440}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{3963}(373,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{53}{660}\right)\) | \(e\left(\frac{71}{264}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{163}{165}\right)\) | \(e\left(\frac{823}{1320}\right)\) | \(e\left(\frac{67}{440}\right)\) | \(e\left(\frac{8}{15}\right)\) |
| \(\chi_{3963}(376,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{229}{660}\right)\) | \(e\left(\frac{247}{264}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{119}{165}\right)\) | \(e\left(\frac{119}{1320}\right)\) | \(e\left(\frac{331}{440}\right)\) | \(e\left(\frac{4}{15}\right)\) |