from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(731, base_ring=CyclotomicField(336))
M = H._module
chi = DirichletCharacter(H, M([21,8]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,731))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(731\) | |
| Conductor: | \(731\) | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{731}(3,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{29}{336}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{305}{336}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{29}{168}\right)\) | \(e\left(\frac{143}{336}\right)\) | \(e\left(\frac{17}{112}\right)\) |
| \(\chi_{731}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{305}{336}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{149}{336}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{137}{168}\right)\) | \(e\left(\frac{299}{336}\right)\) | \(e\left(\frac{5}{112}\right)\) |
| \(\chi_{731}(12,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{41}{336}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{269}{336}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{41}{168}\right)\) | \(e\left(\frac{179}{336}\right)\) | \(e\left(\frac{109}{112}\right)\) |
| \(\chi_{731}(20,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{317}{336}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{113}{336}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{149}{168}\right)\) | \(e\left(\frac{335}{336}\right)\) | \(e\left(\frac{97}{112}\right)\) |
| \(\chi_{731}(28,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{187}{336}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{55}{336}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{19}{168}\right)\) | \(e\left(\frac{169}{336}\right)\) | \(e\left(\frac{71}{112}\right)\) |
| \(\chi_{731}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{265}{336}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{157}{336}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{97}{168}\right)\) | \(e\left(\frac{67}{336}\right)\) | \(e\left(\frac{109}{112}\right)\) |
| \(\chi_{731}(46,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{281}{336}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{221}{336}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{227}{336}\right)\) | \(e\left(\frac{45}{112}\right)\) |
| \(\chi_{731}(48,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{53}{336}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{233}{336}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{215}{336}\right)\) | \(e\left(\frac{89}{112}\right)\) |
| \(\chi_{731}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{295}{336}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{67}{336}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{127}{168}\right)\) | \(e\left(\frac{157}{336}\right)\) | \(e\left(\frac{3}{112}\right)\) |
| \(\chi_{731}(62,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{299}{336}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{167}{336}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{281}{336}\right)\) | \(e\left(\frac{71}{112}\right)\) |
| \(\chi_{731}(63,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{233}{336}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{29}{336}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{65}{168}\right)\) | \(e\left(\frac{83}{336}\right)\) | \(e\left(\frac{13}{112}\right)\) |
| \(\chi_{731}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{61}{336}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{97}{336}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{127}{336}\right)\) | \(e\left(\frac{1}{112}\right)\) |
| \(\chi_{731}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{193}{336}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{37}{336}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{25}{168}\right)\) | \(e\left(\frac{187}{336}\right)\) | \(e\left(\frac{5}{112}\right)\) |
| \(\chi_{731}(91,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{179}{336}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{191}{336}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{11}{168}\right)\) | \(e\left(\frac{257}{336}\right)\) | \(e\left(\frac{47}{112}\right)\) |
| \(\chi_{731}(105,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{173}{336}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{209}{336}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{5}{168}\right)\) | \(e\left(\frac{239}{336}\right)\) | \(e\left(\frac{1}{112}\right)\) |
| \(\chi_{731}(112,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{199}{336}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{19}{336}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{31}{168}\right)\) | \(e\left(\frac{205}{336}\right)\) | \(e\left(\frac{51}{112}\right)\) |
| \(\chi_{731}(114,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{313}{336}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{13}{336}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{145}{168}\right)\) | \(e\left(\frac{211}{336}\right)\) | \(e\left(\frac{29}{112}\right)\) |
| \(\chi_{731}(116,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{277}{336}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{121}{336}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{109}{168}\right)\) | \(e\left(\frac{103}{336}\right)\) | \(e\left(\frac{89}{112}\right)\) |
| \(\chi_{731}(141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{209}{336}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{101}{336}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{41}{168}\right)\) | \(e\left(\frac{11}{336}\right)\) | \(e\left(\frac{53}{112}\right)\) |
| \(\chi_{731}(147,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{43}{336}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{151}{336}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{43}{168}\right)\) | \(e\left(\frac{73}{336}\right)\) | \(e\left(\frac{87}{112}\right)\) |
| \(\chi_{731}(148,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{89}{336}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{125}{336}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{323}{336}\right)\) | \(e\left(\frac{29}{112}\right)\) |
| \(\chi_{731}(158,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{97}{336}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{325}{336}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{97}{168}\right)\) | \(e\left(\frac{235}{336}\right)\) | \(e\left(\frac{53}{112}\right)\) |
| \(\chi_{731}(159,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{67}{336}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{79}{336}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{67}{168}\right)\) | \(e\left(\frac{145}{336}\right)\) | \(e\left(\frac{47}{112}\right)\) |
| \(\chi_{731}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{247}{336}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{211}{336}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{79}{168}\right)\) | \(e\left(\frac{13}{336}\right)\) | \(e\left(\frac{83}{112}\right)\) |
| \(\chi_{731}(175,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{113}{336}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{53}{336}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{59}{336}\right)\) | \(e\left(\frac{101}{112}\right)\) |
| \(\chi_{731}(177,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{95}{336}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{107}{336}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{5}{336}\right)\) | \(e\left(\frac{75}{112}\right)\) |
| \(\chi_{731}(184,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{293}{336}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{185}{336}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{125}{168}\right)\) | \(e\left(\frac{263}{336}\right)\) | \(e\left(\frac{25}{112}\right)\) |
| \(\chi_{731}(190,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{253}{336}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{193}{336}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{85}{168}\right)\) | \(e\left(\frac{31}{336}\right)\) | \(e\left(\frac{17}{112}\right)\) |
| \(\chi_{731}(192,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{65}{336}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{197}{336}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{65}{168}\right)\) | \(e\left(\frac{251}{336}\right)\) | \(e\left(\frac{69}{112}\right)\) |
| \(\chi_{731}(198,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{283}{336}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{103}{336}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{115}{168}\right)\) | \(e\left(\frac{121}{336}\right)\) | \(e\left(\frac{23}{112}\right)\) |
| \(\chi_{731}(201,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{181}{336}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{73}{336}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{151}{336}\right)\) | \(e\left(\frac{25}{112}\right)\) |