from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8352, base_ring=CyclotomicField(168))
M = H._module
chi = DirichletCharacter(H, M([84,105,112,138]))
chi.galois_orbit()
[g,chi] = znchar(Mod(619,8352))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8352\) | |
| Conductor: | \(8352\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(168\) | 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: | $\Q(\zeta_{168})$ |
| Fixed field: | Number field defined by a degree 168 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8352}(619,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{168}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{139}{168}\right)\) | \(e\left(\frac{83}{168}\right)\) | \(-i\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{17}{56}\right)\) |
| \(\chi_{8352}(859,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{168}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{47}{168}\right)\) | \(e\left(\frac{103}{168}\right)\) | \(-i\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{13}{56}\right)\) |
| \(\chi_{8352}(907,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{163}{168}\right)\) | \(e\left(\frac{107}{168}\right)\) | \(-i\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{1}{56}\right)\) |
| \(\chi_{8352}(931,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{168}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{29}{168}\right)\) | \(e\left(\frac{85}{168}\right)\) | \(i\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{39}{56}\right)\) |
| \(\chi_{8352}(1075,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{168}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{17}{168}\right)\) | \(e\left(\frac{73}{168}\right)\) | \(i\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{19}{56}\right)\) |
| \(\chi_{8352}(1123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{168}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{37}{168}\right)\) | \(e\left(\frac{149}{168}\right)\) | \(i\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{15}{56}\right)\) |
| \(\chi_{8352}(1291,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{168}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{11}{168}\right)\) | \(e\left(\frac{67}{168}\right)\) | \(-i\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{9}{56}\right)\) |
| \(\chi_{8352}(1411,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{125}{168}\right)\) | \(i\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{31}{56}\right)\) |
| \(\chi_{8352}(2011,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{168}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{167}{168}\right)\) | \(e\left(\frac{55}{168}\right)\) | \(-i\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{45}{56}\right)\) |
| \(\chi_{8352}(2131,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{168}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{25}{168}\right)\) | \(e\left(\frac{137}{168}\right)\) | \(i\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{51}{56}\right)\) |
| \(\chi_{8352}(2299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{168}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{23}{168}\right)\) | \(e\left(\frac{79}{168}\right)\) | \(-i\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{29}{56}\right)\) |
| \(\chi_{8352}(2347,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{168}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{115}{168}\right)\) | \(e\left(\frac{59}{168}\right)\) | \(-i\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{33}{56}\right)\) |
| \(\chi_{8352}(2491,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{168}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{127}{168}\right)\) | \(e\left(\frac{71}{168}\right)\) | \(-i\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{53}{56}\right)\) |
| \(\chi_{8352}(2515,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{65}{168}\right)\) | \(e\left(\frac{121}{168}\right)\) | \(i\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{43}{56}\right)\) |
| \(\chi_{8352}(2563,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{168}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{5}{168}\right)\) | \(i\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{55}{56}\right)\) |
| \(\chi_{8352}(2803,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{168}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{41}{168}\right)\) | \(e\left(\frac{97}{168}\right)\) | \(i\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{3}{56}\right)\) |
| \(\chi_{8352}(3523,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{168}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{109}{168}\right)\) | \(i\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{23}{56}\right)\) |
| \(\chi_{8352}(3643,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{168}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{103}{168}\right)\) | \(e\left(\frac{47}{168}\right)\) | \(-i\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{13}{56}\right)\) |
| \(\chi_{8352}(3715,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{168}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{85}{168}\right)\) | \(e\left(\frac{29}{168}\right)\) | \(i\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{39}{56}\right)\) |
| \(\chi_{8352}(3739,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{168}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{143}{168}\right)\) | \(e\left(\frac{31}{168}\right)\) | \(-i\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{5}{56}\right)\) |
| \(\chi_{8352}(3859,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{168}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{73}{168}\right)\) | \(e\left(\frac{17}{168}\right)\) | \(i\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{19}{56}\right)\) |
| \(\chi_{8352}(3883,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{168}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{155}{168}\right)\) | \(e\left(\frac{43}{168}\right)\) | \(-i\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{25}{56}\right)\) |
| \(\chi_{8352}(3955,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{168}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{145}{168}\right)\) | \(i\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{27}{56}\right)\) |
| \(\chi_{8352}(4075,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{168}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{67}{168}\right)\) | \(e\left(\frac{11}{168}\right)\) | \(-i\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{9}{56}\right)\) |
| \(\chi_{8352}(4795,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{55}{168}\right)\) | \(e\left(\frac{167}{168}\right)\) | \(-i\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{45}{56}\right)\) |
| \(\chi_{8352}(5035,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{157}{168}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{19}{168}\right)\) | \(-i\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{41}{56}\right)\) |
| \(\chi_{8352}(5083,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{168}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{79}{168}\right)\) | \(e\left(\frac{23}{168}\right)\) | \(-i\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{29}{56}\right)\) |
| \(\chi_{8352}(5107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{168}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{1}{168}\right)\) | \(i\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{11}{56}\right)\) |
| \(\chi_{8352}(5251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{168}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{101}{168}\right)\) | \(e\left(\frac{157}{168}\right)\) | \(i\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{47}{56}\right)\) |
| \(\chi_{8352}(5299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{121}{168}\right)\) | \(e\left(\frac{65}{168}\right)\) | \(i\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{43}{56}\right)\) |
| \(\chi_{8352}(5467,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{151}{168}\right)\) | \(-i\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{37}{56}\right)\) |