from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(731, base_ring=CyclotomicField(56))
M = H._module
chi = DirichletCharacter(H, M([35,32]))
chi.galois_orbit()
[g,chi] = znchar(Mod(59,731))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(731\) | |
| Conductor: | \(731\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(56\) | 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: | $\Q(\zeta_{56})$ |
| Fixed field: | Number field defined by a degree 56 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{731}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{29}{56}\right)\) |
| \(\chi_{731}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{55}{56}\right)\) |
| \(\chi_{731}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{37}{56}\right)\) |
| \(\chi_{731}(145,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{1}{56}\right)\) |
| \(\chi_{731}(213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{9}{56}\right)\) |
| \(\chi_{731}(219,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{11}{56}\right)\) |
| \(\chi_{731}(236,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{19}{56}\right)\) |
| \(\chi_{731}(274,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{15}{56}\right)\) |
| \(\chi_{731}(342,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{23}{56}\right)\) |
| \(\chi_{731}(348,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{53}{56}\right)\) |
| \(\chi_{731}(355,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{3}{56}\right)\) |
| \(\chi_{731}(365,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{5}{56}\right)\) |
| \(\chi_{731}(434,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{25}{56}\right)\) |
| \(\chi_{731}(451,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{33}{56}\right)\) |
| \(\chi_{731}(484,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{45}{56}\right)\) |
| \(\chi_{731}(508,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{27}{56}\right)\) |
| \(\chi_{731}(563,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{39}{56}\right)\) |
| \(\chi_{731}(570,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{17}{56}\right)\) |
| \(\chi_{731}(580,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{47}{56}\right)\) |
| \(\chi_{731}(637,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{13}{56}\right)\) |
| \(\chi_{731}(661,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{43}{56}\right)\) |
| \(\chi_{731}(699,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{31}{56}\right)\) |
| \(\chi_{731}(723,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{41}{56}\right)\) |
| \(\chi_{731}(729,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{51}{56}\right)\) |