from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3136, base_ring=CyclotomicField(168))
M = H._module
chi = DirichletCharacter(H, M([84,147,152]))
chi.galois_orbit()
[g,chi] = znchar(Mod(23,3136))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3136\) | |
| Conductor: | \(1568\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(168\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1568.ci | 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: | $\Q(\zeta_{168})$ |
| Fixed field: | Number field defined by a degree 168 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3136}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{168}\right)\) | \(e\left(\frac{19}{168}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{11}{168}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) |
| \(\chi_{3136}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{168}\right)\) | \(e\left(\frac{101}{168}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{85}{168}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{17}{84}\right)\) |
| \(\chi_{3136}(135,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{168}\right)\) | \(e\left(\frac{121}{168}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{17}{168}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{37}{84}\right)\) |
| \(\chi_{3136}(151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{67}{168}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{47}{84}\right)\) |
| \(\chi_{3136}(247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{168}\right)\) | \(e\left(\frac{55}{168}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{23}{168}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) |
| \(\chi_{3136}(359,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{168}\right)\) | \(e\left(\frac{157}{168}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{29}{168}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{73}{84}\right)\) |
| \(\chi_{3136}(375,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{168}\right)\) | \(e\left(\frac{23}{168}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{31}{168}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) |
| \(\chi_{3136}(487,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{168}\right)\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) |
| \(\chi_{3136}(583,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{25}{168}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{41}{168}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{25}{84}\right)\) |
| \(\chi_{3136}(599,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{83}{168}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{163}{168}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{83}{84}\right)\) |
| \(\chi_{3136}(695,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{127}{168}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{47}{168}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{43}{84}\right)\) |
| \(\chi_{3136}(711,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{168}\right)\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{145}{168}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) |
| \(\chi_{3136}(807,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{61}{84}\right)\) |
| \(\chi_{3136}(823,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{168}\right)\) | \(e\left(\frac{143}{168}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{127}{168}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{59}{84}\right)\) |
| \(\chi_{3136}(919,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{168}\right)\) | \(e\left(\frac{163}{168}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{59}{168}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) |
| \(\chi_{3136}(935,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{168}\right)\) | \(e\left(\frac{5}{168}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{109}{168}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) |
| \(\chi_{3136}(1031,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{168}\right)\) | \(e\left(\frac{97}{168}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{65}{168}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{13}{84}\right)\) |
| \(\chi_{3136}(1143,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{168}\right)\) | \(e\left(\frac{31}{168}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{71}{168}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{31}{84}\right)\) |
| \(\chi_{3136}(1159,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{168}\right)\) | \(e\left(\frac{65}{168}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{73}{168}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{65}{84}\right)\) |
| \(\chi_{3136}(1271,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{168}\right)\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{55}{168}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) |
| \(\chi_{3136}(1367,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{67}{168}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{83}{168}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{67}{84}\right)\) |
| \(\chi_{3136}(1383,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{168}\right)\) | \(e\left(\frac{125}{168}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{37}{168}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) |
| \(\chi_{3136}(1479,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{168}\right)\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) |
| \(\chi_{3136}(1495,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{168}\right)\) | \(e\left(\frac{155}{168}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{19}{168}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{71}{84}\right)\) |
| \(\chi_{3136}(1591,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{103}{168}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) |
| \(\chi_{3136}(1607,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{168}\right)\) | \(e\left(\frac{17}{168}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{17}{84}\right)\) |
| \(\chi_{3136}(1703,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{168}\right)\) | \(e\left(\frac{37}{168}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{101}{168}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{37}{84}\right)\) |
| \(\chi_{3136}(1719,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{168}\right)\) | \(e\left(\frac{47}{168}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{47}{84}\right)\) |
| \(\chi_{3136}(1815,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{168}\right)\) | \(e\left(\frac{139}{168}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{107}{168}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) |
| \(\chi_{3136}(1927,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{168}\right)\) | \(e\left(\frac{73}{168}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{73}{84}\right)\) |
| \(\chi_{3136}(1943,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{168}\right)\) | \(e\left(\frac{107}{168}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{115}{168}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) |