from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4056, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([0,39,0,70]))
chi.galois_orbit()
[g,chi] = znchar(Mod(61,4056))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4056\) | |
| Conductor: | \(1352\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(78\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1352.bn | 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: | $\Q(\zeta_{39})$ |
| Fixed field: | Number field defined by a degree 78 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4056}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{47}{78}\right)\) |
| \(\chi_{4056}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{31}{78}\right)\) |
| \(\chi_{4056}(373,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{59}{78}\right)\) |
| \(\chi_{4056}(445,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{67}{78}\right)\) |
| \(\chi_{4056}(685,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{71}{78}\right)\) |
| \(\chi_{4056}(757,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{25}{78}\right)\) |
| \(\chi_{4056}(997,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{5}{78}\right)\) |
| \(\chi_{4056}(1069,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{61}{78}\right)\) |
| \(\chi_{4056}(1309,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{17}{78}\right)\) |
| \(\chi_{4056}(1381,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{19}{78}\right)\) |
| \(\chi_{4056}(1621,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{29}{78}\right)\) |
| \(\chi_{4056}(1693,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{55}{78}\right)\) |
| \(\chi_{4056}(1933,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{41}{78}\right)\) |
| \(\chi_{4056}(2245,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{53}{78}\right)\) |
| \(\chi_{4056}(2317,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{49}{78}\right)\) |
| \(\chi_{4056}(2629,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{7}{78}\right)\) |
| \(\chi_{4056}(2869,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{77}{78}\right)\) |
| \(\chi_{4056}(2941,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{43}{78}\right)\) |
| \(\chi_{4056}(3181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{11}{78}\right)\) |
| \(\chi_{4056}(3253,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{78}\right)\) |
| \(\chi_{4056}(3493,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{23}{78}\right)\) |
| \(\chi_{4056}(3565,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{37}{78}\right)\) |
| \(\chi_{4056}(3805,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{35}{78}\right)\) |
| \(\chi_{4056}(3877,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{73}{78}\right)\) |