from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(361, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([26]))
chi.galois_orbit()
[g,chi] = znchar(Mod(20,361))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(361\) | |
| Conductor: | \(361\) | 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: | 19.19.10842505080063916320800450434338728415281531281.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{361}(20,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) |
| \(\chi_{361}(39,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) |
| \(\chi_{361}(58,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) |
| \(\chi_{361}(77,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) |
| \(\chi_{361}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) |
| \(\chi_{361}(115,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) |
| \(\chi_{361}(134,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) |
| \(\chi_{361}(153,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) |
| \(\chi_{361}(172,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) |
| \(\chi_{361}(191,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) |
| \(\chi_{361}(210,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) |
| \(\chi_{361}(229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) |
| \(\chi_{361}(248,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) |
| \(\chi_{361}(267,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) |
| \(\chi_{361}(286,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) |
| \(\chi_{361}(305,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) |
| \(\chi_{361}(324,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) |
| \(\chi_{361}(343,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) |