from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1666, base_ring=CyclotomicField(56))
M = H._module
chi = DirichletCharacter(H, M([40,21]))
chi.galois_orbit()
[g,chi] = znchar(Mod(15,1666))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1666\) | |
| Conductor: | \(833\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(56\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 833.bf | 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1666}(15,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(i\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{15}{56}\right)\) |
| \(\chi_{1666}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(-i\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{45}{56}\right)\) |
| \(\chi_{1666}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(-i\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{9}{56}\right)\) |
| \(\chi_{1666}(155,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(i\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{11}{56}\right)\) |
| \(\chi_{1666}(253,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(i\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{39}{56}\right)\) |
| \(\chi_{1666}(281,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(-i\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{13}{56}\right)\) |
| \(\chi_{1666}(365,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(-i\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{33}{56}\right)\) |
| \(\chi_{1666}(519,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(-i\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{37}{56}\right)\) |
| \(\chi_{1666}(603,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(-i\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{1}{56}\right)\) |
| \(\chi_{1666}(631,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(i\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{3}{56}\right)\) |
| \(\chi_{1666}(729,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(i\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{31}{56}\right)\) |
| \(\chi_{1666}(757,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(-i\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{5}{56}\right)\) |
| \(\chi_{1666}(841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(-i\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{25}{56}\right)\) |
| \(\chi_{1666}(869,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(i\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{27}{56}\right)\) |
| \(\chi_{1666}(967,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(i\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{55}{56}\right)\) |
| \(\chi_{1666}(995,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(-i\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{29}{56}\right)\) |
| \(\chi_{1666}(1107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(i\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{51}{56}\right)\) |
| \(\chi_{1666}(1205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(i\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{23}{56}\right)\) |
| \(\chi_{1666}(1233,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(-i\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{53}{56}\right)\) |
| \(\chi_{1666}(1317,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(-i\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{17}{56}\right)\) |
| \(\chi_{1666}(1345,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(i\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{19}{56}\right)\) |
| \(\chi_{1666}(1443,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(i\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{47}{56}\right)\) |
| \(\chi_{1666}(1555,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(-i\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{41}{56}\right)\) |
| \(\chi_{1666}(1583,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(i\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{43}{56}\right)\) |