from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7098, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([0,0,38]))
chi.galois_orbit()
[g,chi] = znchar(Mod(211,7098))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(7098\) | |
Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(39\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 169.i | 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 39 polynomial |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(5\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{7098}(211,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) |
\(\chi_{7098}(295,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) |
\(\chi_{7098}(757,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) |
\(\chi_{7098}(841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) |
\(\chi_{7098}(1303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) |
\(\chi_{7098}(1387,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) |
\(\chi_{7098}(1849,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) |
\(\chi_{7098}(1933,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) |
\(\chi_{7098}(2395,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) |
\(\chi_{7098}(2479,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) |
\(\chi_{7098}(2941,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) |
\(\chi_{7098}(3025,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) |
\(\chi_{7098}(3487,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) |
\(\chi_{7098}(4117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) |
\(\chi_{7098}(4579,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) |
\(\chi_{7098}(4663,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) |
\(\chi_{7098}(5125,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) |
\(\chi_{7098}(5209,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) |
\(\chi_{7098}(5671,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) |
\(\chi_{7098}(5755,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) |
\(\chi_{7098}(6217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) |
\(\chi_{7098}(6301,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) |
\(\chi_{7098}(6763,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) |
\(\chi_{7098}(6847,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) |