from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(869856, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([0,60,40,40,25,14]))
chi.galois_orbit()
[g,chi] = znchar(Mod(20201,869856))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(869856\) | |
| Conductor: | \(434928\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(80\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 434928.ivz | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{80})$ |
| Fixed field: | Number field defined by a degree 80 polynomial |
First 31 of 32 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{869856}(20201,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{13}{80}\right)\) |
| \(\chi_{869856}(41417,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{61}{80}\right)\) |
| \(\chi_{869856}(62009,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{27}{80}\right)\) |
| \(\chi_{869856}(110681,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{31}{80}\right)\) |
| \(\chi_{869856}(149369,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{19}{80}\right)\) |
| \(\chi_{869856}(212393,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{69}{80}\right)\) |
| \(\chi_{869856}(216761,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{79}{80}\right)\) |
| \(\chi_{869856}(255449,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{51}{80}\right)\) |
| \(\chi_{869856}(276041,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{53}{80}\right)\) |
| \(\chi_{869856}(296009,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{77}{80}\right)\) |
| \(\chi_{869856}(297257,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{21}{80}\right)\) |
| \(\chi_{869856}(325961,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{57}{80}\right)\) |
| \(\chi_{869856}(355289,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{7}{80}\right)\) |
| \(\chi_{869856}(446393,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{3}{80}\right)\) |
| \(\chi_{869856}(479465,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{17}{80}\right)\) |
| \(\chi_{869856}(482585,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{23}{80}\right)\) |
| \(\chi_{869856}(507545,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{59}{80}\right)\) |
| \(\chi_{869856}(551849,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{37}{80}\right)\) |
| \(\chi_{869856}(573689,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{67}{80}\right)\) |
| \(\chi_{869856}(580553,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{41}{80}\right)\) |
| \(\chi_{869856}(601769,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{73}{80}\right)\) |
| \(\chi_{869856}(613625,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{11}{80}\right)\) |
| \(\chi_{869856}(662297,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{47}{80}\right)\) |
| \(\chi_{869856}(665417,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{9}{80}\right)\) |
| \(\chi_{869856}(673529,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{71}{80}\right)\) |
| \(\chi_{869856}(734057,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{1}{80}\right)\) |
| \(\chi_{869856}(755273,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{33}{80}\right)\) |
| \(\chi_{869856}(779609,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{39}{80}\right)\) |
| \(\chi_{869856}(789593,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{63}{80}\right)\) |
| \(\chi_{869856}(804569,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{43}{80}\right)\) |
| \(\chi_{869856}(818921,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{49}{80}\right)\) |