from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4056, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([0,39,39,20]))
chi.galois_orbit()
[g,chi] = znchar(Mod(29,4056))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(4056\) | |
Conductor: | \(4056\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(78\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | 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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{4056}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{29}{39}\right)\) |
\(\chi_{4056}(269,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{34}{39}\right)\) |
\(\chi_{4056}(341,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{8}{39}\right)\) |
\(\chi_{4056}(581,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) |
\(\chi_{4056}(893,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) |
\(\chi_{4056}(965,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) |
\(\chi_{4056}(1277,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) |
\(\chi_{4056}(1517,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{19}{39}\right)\) |
\(\chi_{4056}(1589,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{2}{39}\right)\) |
\(\chi_{4056}(1829,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{25}{39}\right)\) |
\(\chi_{4056}(1901,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) |
\(\chi_{4056}(2141,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{31}{39}\right)\) |
\(\chi_{4056}(2213,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{38}{39}\right)\) |
\(\chi_{4056}(2453,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) |
\(\chi_{4056}(2525,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{17}{39}\right)\) |
\(\chi_{4056}(2765,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) |
\(\chi_{4056}(2837,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{35}{39}\right)\) |
\(\chi_{4056}(3077,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) |
\(\chi_{4056}(3149,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{14}{39}\right)\) |
\(\chi_{4056}(3389,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{16}{39}\right)\) |
\(\chi_{4056}(3461,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) |
\(\chi_{4056}(3701,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{22}{39}\right)\) |
\(\chi_{4056}(3773,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) |
\(\chi_{4056}(4013,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) |