from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8352, base_ring=CyclotomicField(56))
M = H._module
chi = DirichletCharacter(H, M([28,35,28,48]))
chi.galois_orbit()
[g,chi] = znchar(Mod(107,8352))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8352\) | |
| Conductor: | \(2784\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(56\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2784.eh | 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: | $\Q(\zeta_{56})$ |
| Fixed field: | Number field defined by a degree 56 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8352}(107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(1\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{56}\right)\) |
| \(\chi_{8352}(683,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(1\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{41}{56}\right)\) |
| \(\chi_{8352}(1475,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(1\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{31}{56}\right)\) |
| \(\chi_{8352}(1619,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(1\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{56}\right)\) |
| \(\chi_{8352}(1763,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(1\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{23}{56}\right)\) |
| \(\chi_{8352}(1979,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(1\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{53}{56}\right)\) |
| \(\chi_{8352}(2195,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(1\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{43}{56}\right)\) |
| \(\chi_{8352}(2771,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(1\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{27}{56}\right)\) |
| \(\chi_{8352}(3563,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(1\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{17}{56}\right)\) |
| \(\chi_{8352}(3707,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(1\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{56}\right)\) |
| \(\chi_{8352}(3851,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(1\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{9}{56}\right)\) |
| \(\chi_{8352}(4067,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(1\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{39}{56}\right)\) |
| \(\chi_{8352}(4283,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(1\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{29}{56}\right)\) |
| \(\chi_{8352}(4859,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(1\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{56}\right)\) |
| \(\chi_{8352}(5651,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(1\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{56}\right)\) |
| \(\chi_{8352}(5795,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(1\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{47}{56}\right)\) |
| \(\chi_{8352}(5939,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(1\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{51}{56}\right)\) |
| \(\chi_{8352}(6155,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(1\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{25}{56}\right)\) |
| \(\chi_{8352}(6371,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(1\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{15}{56}\right)\) |
| \(\chi_{8352}(6947,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(1\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{55}{56}\right)\) |
| \(\chi_{8352}(7739,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(1\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{45}{56}\right)\) |
| \(\chi_{8352}(7883,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(1\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{33}{56}\right)\) |
| \(\chi_{8352}(8027,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(1\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{37}{56}\right)\) |
| \(\chi_{8352}(8243,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(1\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{11}{56}\right)\) |