from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, base_ring=CyclotomicField(336))
M = H._module
chi = DirichletCharacter(H, M([0,273,104]))
chi.galois_orbit()
[g,chi] = znchar(Mod(12,8041))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8041\) | |
| Conductor: | \(731\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(336\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 731.bm | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8041}(12,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{41}{336}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{269}{336}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{41}{168}\right)\) | \(e\left(\frac{179}{336}\right)\) | \(e\left(\frac{197}{336}\right)\) |
| \(\chi_{8041}(177,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{95}{336}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{107}{336}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{5}{336}\right)\) | \(e\left(\frac{227}{336}\right)\) |
| \(\chi_{8041}(243,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{145}{336}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{181}{336}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{145}{168}\right)\) | \(e\left(\frac{43}{336}\right)\) | \(e\left(\frac{205}{336}\right)\) |
| \(\chi_{8041}(320,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{5}{336}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{41}{336}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{5}{168}\right)\) | \(e\left(\frac{71}{336}\right)\) | \(e\left(\frac{65}{336}\right)\) |
| \(\chi_{8041}(364,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{191}{336}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{155}{336}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{23}{168}\right)\) | \(e\left(\frac{293}{336}\right)\) | \(e\left(\frac{131}{336}\right)\) |
| \(\chi_{8041}(507,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{37}{336}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{169}{336}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{37}{168}\right)\) | \(e\left(\frac{55}{336}\right)\) | \(e\left(\frac{145}{336}\right)\) |
| \(\chi_{8041}(760,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{265}{336}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{157}{336}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{97}{168}\right)\) | \(e\left(\frac{67}{336}\right)\) | \(e\left(\frac{85}{336}\right)\) |
| \(\chi_{8041}(793,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{299}{336}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{167}{336}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{281}{336}\right)\) | \(e\left(\frac{191}{336}\right)\) |
| \(\chi_{8041}(804,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{193}{336}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{37}{336}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{25}{168}\right)\) | \(e\left(\frac{187}{336}\right)\) | \(e\left(\frac{157}{336}\right)\) |
| \(\chi_{8041}(958,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{83}{336}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{143}{336}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{83}{168}\right)\) | \(e\left(\frac{305}{336}\right)\) | \(e\left(\frac{71}{336}\right)\) |
| \(\chi_{8041}(980,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{331}{336}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{295}{336}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{163}{168}\right)\) | \(e\left(\frac{265}{336}\right)\) | \(e\left(\frac{271}{336}\right)\) |
| \(\chi_{8041}(1431,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{125}{336}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{17}{336}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{125}{168}\right)\) | \(e\left(\frac{95}{336}\right)\) | \(e\left(\frac{281}{336}\right)\) |
| \(\chi_{8041}(1508,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{281}{336}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{221}{336}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{227}{336}\right)\) | \(e\left(\frac{293}{336}\right)\) |
| \(\chi_{8041}(1574,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{199}{336}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{19}{336}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{31}{168}\right)\) | \(e\left(\frac{205}{336}\right)\) | \(e\left(\frac{235}{336}\right)\) |
| \(\chi_{8041}(1695,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{169}{336}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{109}{336}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{115}{336}\right)\) | \(e\left(\frac{181}{336}\right)\) |
| \(\chi_{8041}(1706,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{307}{336}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{31}{336}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{139}{168}\right)\) | \(e\left(\frac{193}{336}\right)\) | \(e\left(\frac{295}{336}\right)\) |
| \(\chi_{8041}(1739,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{257}{336}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{293}{336}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{155}{336}\right)\) | \(e\left(\frac{317}{336}\right)\) |
| \(\chi_{8041}(1882,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{185}{336}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{173}{336}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{17}{168}\right)\) | \(e\left(\frac{275}{336}\right)\) | \(e\left(\frac{53}{336}\right)\) |
| \(\chi_{8041}(1926,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{289}{336}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{85}{336}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{121}{168}\right)\) | \(e\left(\frac{139}{336}\right)\) | \(e\left(\frac{61}{336}\right)\) |
| \(\chi_{8041}(2047,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{31}{336}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{187}{336}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{31}{168}\right)\) | \(e\left(\frac{37}{336}\right)\) | \(e\left(\frac{67}{336}\right)\) |
| \(\chi_{8041}(2069,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{137}{336}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{317}{336}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{137}{168}\right)\) | \(e\left(\frac{131}{336}\right)\) | \(e\left(\frac{101}{336}\right)\) |
| \(\chi_{8041}(2135,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{103}{336}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{307}{336}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{103}{168}\right)\) | \(e\left(\frac{253}{336}\right)\) | \(e\left(\frac{331}{336}\right)\) |
| \(\chi_{8041}(2179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{13}{336}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{241}{336}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{319}{336}\right)\) | \(e\left(\frac{169}{336}\right)\) |
| \(\chi_{8041}(2256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{233}{336}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{29}{336}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{65}{168}\right)\) | \(e\left(\frac{83}{336}\right)\) | \(e\left(\frac{5}{336}\right)\) |
| \(\chi_{8041}(2377,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{293}{336}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{185}{336}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{125}{168}\right)\) | \(e\left(\frac{263}{336}\right)\) | \(e\left(\frac{113}{336}\right)\) |
| \(\chi_{8041}(2454,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{323}{336}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{95}{336}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{155}{168}\right)\) | \(e\left(\frac{17}{336}\right)\) | \(e\left(\frac{167}{336}\right)\) |
| \(\chi_{8041}(2608,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{271}{336}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{139}{336}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{103}{168}\right)\) | \(e\left(\frac{85}{336}\right)\) | \(e\left(\frac{163}{336}\right)\) |
| \(\chi_{8041}(2641,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{211}{336}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{319}{336}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{43}{168}\right)\) | \(e\left(\frac{241}{336}\right)\) | \(e\left(\frac{55}{336}\right)\) |
| \(\chi_{8041}(2696,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{151}{336}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{163}{336}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{61}{336}\right)\) | \(e\left(\frac{283}{336}\right)\) |
| \(\chi_{8041}(2828,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{227}{336}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{47}{336}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{59}{168}\right)\) | \(e\left(\frac{65}{336}\right)\) | \(e\left(\frac{263}{336}\right)\) |
| \(\chi_{8041}(2850,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{251}{336}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{311}{336}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{83}{168}\right)\) | \(e\left(\frac{137}{336}\right)\) | \(e\left(\frac{239}{336}\right)\) |