from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1011, base_ring=CyclotomicField(336))
M = H._module
chi = DirichletCharacter(H, M([168,305]))
chi.galois_orbit()
[g,chi] = znchar(Mod(20,1011))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1011\) | |
| Conductor: | \(1011\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(336\) | 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: | $\Q(\zeta_{336})$ |
| Fixed field: | Number field defined by a degree 336 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1011}(20,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{51}{112}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{305}{336}\right)\) | \(e\left(\frac{81}{112}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{168}\right)\) | \(e\left(\frac{17}{21}\right)\) |
| \(\chi_{1011}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{111}{112}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{229}{336}\right)\) | \(e\left(\frac{5}{112}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{71}{168}\right)\) | \(e\left(\frac{16}{21}\right)\) |
| \(\chi_{1011}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{83}{112}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{257}{336}\right)\) | \(e\left(\frac{33}{112}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{43}{168}\right)\) | \(e\left(\frac{2}{21}\right)\) |
| \(\chi_{1011}(44,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{112}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{107}{336}\right)\) | \(e\left(\frac{107}{112}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{97}{168}\right)\) | \(e\left(\frac{5}{21}\right)\) |
| \(\chi_{1011}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{112}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{331}{336}\right)\) | \(e\left(\frac{107}{112}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{41}{168}\right)\) | \(e\left(\frac{19}{21}\right)\) |
| \(\chi_{1011}(68,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{11}{112}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{281}{336}\right)\) | \(e\left(\frac{57}{112}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{115}{168}\right)\) | \(e\left(\frac{20}{21}\right)\) |
| \(\chi_{1011}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{89}{112}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{227}{336}\right)\) | \(e\left(\frac{3}{112}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{121}{168}\right)\) | \(e\left(\frac{11}{21}\right)\) |
| \(\chi_{1011}(80,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{19}{112}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{241}{336}\right)\) | \(e\left(\frac{17}{112}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{107}{168}\right)\) | \(e\left(\frac{4}{21}\right)\) |
| \(\chi_{1011}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{59}{112}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{265}{336}\right)\) | \(e\left(\frac{41}{112}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{11}{168}\right)\) | \(e\left(\frac{1}{21}\right)\) |
| \(\chi_{1011}(89,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{55}{112}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{173}{336}\right)\) | \(e\left(\frac{61}{112}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{127}{168}\right)\) | \(e\left(\frac{2}{21}\right)\) |
| \(\chi_{1011}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{101}{112}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{167}{336}\right)\) | \(e\left(\frac{55}{112}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{109}{168}\right)\) | \(e\left(\frac{8}{21}\right)\) |
| \(\chi_{1011}(116,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{51}{112}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{193}{336}\right)\) | \(e\left(\frac{81}{112}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{10}{21}\right)\) |
| \(\chi_{1011}(134,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{53}{112}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{71}{336}\right)\) | \(e\left(\frac{71}{112}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{157}{168}\right)\) | \(e\left(\frac{20}{21}\right)\) |
| \(\chi_{1011}(143,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{97}{112}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{187}{336}\right)\) | \(e\left(\frac{75}{112}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{16}{21}\right)\) |
| \(\chi_{1011}(152,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{87}{112}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{336}\right)\) | \(e\left(\frac{13}{112}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{1}{21}\right)\) |
| \(\chi_{1011}(161,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{25}{112}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{211}{336}\right)\) | \(e\left(\frac{99}{112}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{17}{168}\right)\) | \(e\left(\frac{13}{21}\right)\) |
| \(\chi_{1011}(176,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{81}{112}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{43}{336}\right)\) | \(e\left(\frac{43}{112}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{17}{168}\right)\) | \(e\left(\frac{13}{21}\right)\) |
| \(\chi_{1011}(185,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{31}{112}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{181}{336}\right)\) | \(e\left(\frac{69}{112}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{1}{21}\right)\) |
| \(\chi_{1011}(194,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{41}{112}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{19}{336}\right)\) | \(e\left(\frac{19}{112}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{16}{21}\right)\) |
| \(\chi_{1011}(203,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{109}{112}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{239}{336}\right)\) | \(e\left(\frac{15}{112}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{157}{168}\right)\) | \(e\left(\frac{20}{21}\right)\) |
| \(\chi_{1011}(221,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{107}{112}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{25}{336}\right)\) | \(e\left(\frac{25}{112}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{10}{21}\right)\) |
| \(\chi_{1011}(236,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{45}{112}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{335}{336}\right)\) | \(e\left(\frac{111}{112}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{109}{168}\right)\) | \(e\left(\frac{8}{21}\right)\) |
| \(\chi_{1011}(248,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{111}{112}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{336}\right)\) | \(e\left(\frac{5}{112}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{127}{168}\right)\) | \(e\left(\frac{2}{21}\right)\) |
| \(\chi_{1011}(254,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{3}{112}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{97}{336}\right)\) | \(e\left(\frac{97}{112}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{11}{168}\right)\) | \(e\left(\frac{1}{21}\right)\) |
| \(\chi_{1011}(257,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{75}{112}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{73}{336}\right)\) | \(e\left(\frac{73}{112}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{107}{168}\right)\) | \(e\left(\frac{4}{21}\right)\) |
| \(\chi_{1011}(266,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{33}{112}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{59}{336}\right)\) | \(e\left(\frac{59}{112}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{121}{168}\right)\) | \(e\left(\frac{11}{21}\right)\) |
| \(\chi_{1011}(269,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{67}{112}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{113}{336}\right)\) | \(e\left(\frac{1}{112}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{115}{168}\right)\) | \(e\left(\frac{20}{21}\right)\) |
| \(\chi_{1011}(284,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{57}{112}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{163}{336}\right)\) | \(e\left(\frac{51}{112}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{41}{168}\right)\) | \(e\left(\frac{19}{21}\right)\) |
| \(\chi_{1011}(293,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{57}{112}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{275}{336}\right)\) | \(e\left(\frac{51}{112}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{97}{168}\right)\) | \(e\left(\frac{5}{21}\right)\) |
| \(\chi_{1011}(308,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{27}{112}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{89}{336}\right)\) | \(e\left(\frac{89}{112}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{43}{168}\right)\) | \(e\left(\frac{2}{21}\right)\) |
| \(\chi_{1011}(314,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{55}{112}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{61}{336}\right)\) | \(e\left(\frac{61}{112}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{71}{168}\right)\) | \(e\left(\frac{16}{21}\right)\) |