from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6422, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([29,52]))
chi.galois_orbit()
[g,chi] = znchar(Mod(49,6422))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(6422\) | |
Conductor: | \(3211\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(78\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 3211.cq | 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{6422}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{39}\right)\) |
\(\chi_{6422}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{39}\right)\) |
\(\chi_{6422}(543,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{39}\right)\) |
\(\chi_{6422}(615,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{32}{39}\right)\) |
\(\chi_{6422}(1109,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{39}\right)\) |
\(\chi_{6422}(1531,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{39}\right)\) |
\(\chi_{6422}(1603,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{23}{39}\right)\) |
\(\chi_{6422}(2025,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{25}{39}\right)\) |
\(\chi_{6422}(2097,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{38}{39}\right)\) |
\(\chi_{6422}(2519,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{31}{39}\right)\) |
\(\chi_{6422}(2591,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{14}{39}\right)\) |
\(\chi_{6422}(3013,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{37}{39}\right)\) |
\(\chi_{6422}(3085,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{29}{39}\right)\) |
\(\chi_{6422}(3507,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{39}\right)\) |
\(\chi_{6422}(3579,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{39}\right)\) |
\(\chi_{6422}(4001,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{10}{39}\right)\) |
\(\chi_{6422}(4073,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{20}{39}\right)\) |
\(\chi_{6422}(4495,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{16}{39}\right)\) |
\(\chi_{6422}(4567,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{35}{39}\right)\) |
\(\chi_{6422}(4989,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{22}{39}\right)\) |
\(\chi_{6422}(5061,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{39}\right)\) |
\(\chi_{6422}(5483,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{28}{39}\right)\) |
\(\chi_{6422}(5977,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{34}{39}\right)\) |
\(\chi_{6422}(6049,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{39}\right)\) |