from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8352, base_ring=CyclotomicField(168))
M = H._module
chi = DirichletCharacter(H, M([84,105,28,60]))
chi.galois_orbit()
[g,chi] = znchar(Mod(299,8352))
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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8352}(299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{121}{168}\right)\) | \(e\left(\frac{23}{168}\right)\) | \(-1\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{1}{56}\right)\) |
| \(\chi_{8352}(419,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{168}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{47}{168}\right)\) | \(e\left(\frac{145}{168}\right)\) | \(-1\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{55}{56}\right)\) |
| \(\chi_{8352}(515,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{168}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{127}{168}\right)\) | \(e\left(\frac{113}{168}\right)\) | \(-1\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{39}{56}\right)\) |
| \(\chi_{8352}(731,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{109}{168}\right)\) | \(e\left(\frac{11}{168}\right)\) | \(-1\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{37}{56}\right)\) |
| \(\chi_{8352}(875,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{168}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{73}{168}\right)\) | \(e\left(\frac{143}{168}\right)\) | \(-1\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{33}{56}\right)\) |
| \(\chi_{8352}(995,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{168}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{23}{168}\right)\) | \(e\left(\frac{121}{168}\right)\) | \(-1\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{15}{56}\right)\) |
| \(\chi_{8352}(1019,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{83}{168}\right)\) | \(-1\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{45}{56}\right)\) |
| \(\chi_{8352}(1211,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{168}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{29}{168}\right)\) | \(e\left(\frac{43}{168}\right)\) | \(-1\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{53}{56}\right)\) |
| \(\chi_{8352}(1427,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{168}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{11}{168}\right)\) | \(e\left(\frac{109}{168}\right)\) | \(-1\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{51}{56}\right)\) |
| \(\chi_{8352}(1571,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{168}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{143}{168}\right)\) | \(e\left(\frac{73}{168}\right)\) | \(-1\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{47}{56}\right)\) |
| \(\chi_{8352}(1715,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{168}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{83}{168}\right)\) | \(e\left(\frac{13}{168}\right)\) | \(-1\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{3}{56}\right)\) |
| \(\chi_{8352}(1811,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{168}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{19}{168}\right)\) | \(e\left(\frac{5}{168}\right)\) | \(-1\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{27}{56}\right)\) |
| \(\chi_{8352}(2387,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{163}{168}\right)\) | \(e\left(\frac{149}{168}\right)\) | \(-1\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{43}{56}\right)\) |
| \(\chi_{8352}(2507,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{157}{168}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{103}{168}\right)\) | \(-1\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{41}{56}\right)\) |
| \(\chi_{8352}(2603,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{168}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{71}{168}\right)\) | \(-1\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{25}{56}\right)\) |
| \(\chi_{8352}(2819,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{155}{168}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{137}{168}\right)\) | \(-1\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{23}{56}\right)\) |
| \(\chi_{8352}(2963,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{168}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{115}{168}\right)\) | \(e\left(\frac{101}{168}\right)\) | \(-1\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{19}{56}\right)\) |
| \(\chi_{8352}(3083,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{168}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{65}{168}\right)\) | \(e\left(\frac{79}{168}\right)\) | \(-1\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{56}\right)\) |
| \(\chi_{8352}(3107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{55}{168}\right)\) | \(e\left(\frac{41}{168}\right)\) | \(-1\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{31}{56}\right)\) |
| \(\chi_{8352}(3299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{168}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{71}{168}\right)\) | \(e\left(\frac{1}{168}\right)\) | \(-1\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{39}{56}\right)\) |
| \(\chi_{8352}(3515,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{67}{168}\right)\) | \(-1\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{37}{56}\right)\) |
| \(\chi_{8352}(3659,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{17}{168}\right)\) | \(e\left(\frac{31}{168}\right)\) | \(-1\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{33}{56}\right)\) |
| \(\chi_{8352}(3803,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{168}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{125}{168}\right)\) | \(e\left(\frac{139}{168}\right)\) | \(-1\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{45}{56}\right)\) |
| \(\chi_{8352}(3899,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{168}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(-1\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{13}{56}\right)\) |
| \(\chi_{8352}(4475,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{168}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{37}{168}\right)\) | \(e\left(\frac{107}{168}\right)\) | \(-1\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{29}{56}\right)\) |
| \(\chi_{8352}(4595,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{168}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(-1\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{27}{56}\right)\) |
| \(\chi_{8352}(4691,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{168}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{43}{168}\right)\) | \(e\left(\frac{29}{168}\right)\) | \(-1\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{11}{56}\right)\) |
| \(\chi_{8352}(4907,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{168}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{25}{168}\right)\) | \(e\left(\frac{95}{168}\right)\) | \(-1\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{9}{56}\right)\) |
| \(\chi_{8352}(5051,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{168}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{157}{168}\right)\) | \(e\left(\frac{59}{168}\right)\) | \(-1\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{5}{56}\right)\) |
| \(\chi_{8352}(5171,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{107}{168}\right)\) | \(e\left(\frac{37}{168}\right)\) | \(-1\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{43}{56}\right)\) |
| \(\chi_{8352}(5195,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{168}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{97}{168}\right)\) | \(e\left(\frac{167}{168}\right)\) | \(-1\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{17}{56}\right)\) |