from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2366, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([26,46]))
chi.galois_orbit()
[g,chi] = znchar(Mod(9,2366))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(2366\) | |
Conductor: | \(1183\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(39\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 1183.bj | 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: | 39.39.253721406991290895924770503111827676888647505643788160579278763946491805765758913183386148271215912203961.1 |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{2366}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) |
\(\chi_{2366}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) |
\(\chi_{2366}(263,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) |
\(\chi_{2366}(373,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) |
\(\chi_{2366}(445,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) |
\(\chi_{2366}(555,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) |
\(\chi_{2366}(627,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) |
\(\chi_{2366}(737,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) |
\(\chi_{2366}(809,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) |
\(\chi_{2366}(919,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) |
\(\chi_{2366}(1101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) |
\(\chi_{2366}(1173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) |
\(\chi_{2366}(1283,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) |
\(\chi_{2366}(1355,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) |
\(\chi_{2366}(1465,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) |
\(\chi_{2366}(1537,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) |
\(\chi_{2366}(1647,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) |
\(\chi_{2366}(1719,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) |
\(\chi_{2366}(1829,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) |
\(\chi_{2366}(1901,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) |
\(\chi_{2366}(2011,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) |
\(\chi_{2366}(2083,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) |
\(\chi_{2366}(2193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) |
\(\chi_{2366}(2265,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) |