from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1521, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([52,2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(16,1521))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1521\) | |
| Conductor: | \(1521\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(39\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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 | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1521}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{29}{39}\right)\) |
| \(\chi_{1521}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{38}{39}\right)\) |
| \(\chi_{1521}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{16}{39}\right)\) |
| \(\chi_{1521}(250,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{8}{39}\right)\) |
| \(\chi_{1521}(256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{19}{39}\right)\) |
| \(\chi_{1521}(367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{17}{39}\right)\) |
| \(\chi_{1521}(373,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{22}{39}\right)\) |
| \(\chi_{1521}(490,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{25}{39}\right)\) |
| \(\chi_{1521}(601,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{35}{39}\right)\) |
| \(\chi_{1521}(607,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) |
| \(\chi_{1521}(718,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) |
| \(\chi_{1521}(724,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{31}{39}\right)\) |
| \(\chi_{1521}(835,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{14}{39}\right)\) |
| \(\chi_{1521}(841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{34}{39}\right)\) |
| \(\chi_{1521}(952,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) |
| \(\chi_{1521}(958,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) |
| \(\chi_{1521}(1069,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) |
| \(\chi_{1521}(1075,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) |
| \(\chi_{1521}(1186,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{2}{39}\right)\) |
| \(\chi_{1521}(1192,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) |
| \(\chi_{1521}(1303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) |
| \(\chi_{1521}(1309,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) |
| \(\chi_{1521}(1420,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) |
| \(\chi_{1521}(1426,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) |