from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7840, base_ring=CyclotomicField(56))
M = H._module
chi = DirichletCharacter(H, M([28,35,28,20]))
chi.galois_orbit()
[g,chi] = znchar(Mod(139,7840))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(7840\) | |
| Conductor: | \(7840\) | 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 | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{7840}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(1\) |
| \(\chi_{7840}(419,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(1\) |
| \(\chi_{7840}(699,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(1\) |
| \(\chi_{7840}(1259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(1\) |
| \(\chi_{7840}(1539,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(1\) |
| \(\chi_{7840}(1819,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(1\) |
| \(\chi_{7840}(2099,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(1\) |
| \(\chi_{7840}(2379,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(1\) |
| \(\chi_{7840}(2659,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(1\) |
| \(\chi_{7840}(3219,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(1\) |
| \(\chi_{7840}(3499,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(1\) |
| \(\chi_{7840}(3779,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(1\) |
| \(\chi_{7840}(4059,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(1\) |
| \(\chi_{7840}(4339,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(1\) |
| \(\chi_{7840}(4619,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(1\) |
| \(\chi_{7840}(5179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(1\) |
| \(\chi_{7840}(5459,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(1\) |
| \(\chi_{7840}(5739,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(1\) |
| \(\chi_{7840}(6019,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(1\) |
| \(\chi_{7840}(6299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(1\) |
| \(\chi_{7840}(6579,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(1\) |
| \(\chi_{7840}(7139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(1\) |
| \(\chi_{7840}(7419,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(1\) |
| \(\chi_{7840}(7699,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(1\) |