from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(457, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([16]))
chi.galois_orbit()
[g,chi] = znchar(Mod(16,457))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(457\) | |
| Conductor: | \(457\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(19\) | 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: | \(\Q(\zeta_{19})\) |
| Fixed field: | Number field defined by a degree 19 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{457}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) |
| \(\chi_{457}(42,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) |
| \(\chi_{457}(54,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) |
| \(\chi_{457}(68,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) |
| \(\chi_{457}(114,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) |
| \(\chi_{457}(174,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) |
| \(\chi_{457}(185,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) |
| \(\chi_{457}(200,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) |
| \(\chi_{457}(215,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) |
| \(\chi_{457}(218,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) |
| \(\chi_{457}(241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) |
| \(\chi_{457}(256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) |
| \(\chi_{457}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) |
| \(\chi_{457}(347,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) |
| \(\chi_{457}(393,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) |
| \(\chi_{457}(407,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) |
| \(\chi_{457}(440,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) |
| \(\chi_{457}(453,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) |