from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2601, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([0,12]))
chi.galois_orbit()
[g,chi] = znchar(Mod(154,2601))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2601\) | |
| Conductor: | \(289\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(17\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 289.f | 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: | \(\Q(\zeta_{17})\) |
| Fixed field: | Number field defined by a degree 17 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2601}(154,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) |
| \(\chi_{2601}(307,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) |
| \(\chi_{2601}(460,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) |
| \(\chi_{2601}(613,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) |
| \(\chi_{2601}(766,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) |
| \(\chi_{2601}(919,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) |
| \(\chi_{2601}(1072,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) |
| \(\chi_{2601}(1225,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) |
| \(\chi_{2601}(1378,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) |
| \(\chi_{2601}(1531,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) |
| \(\chi_{2601}(1684,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) |
| \(\chi_{2601}(1837,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) |
| \(\chi_{2601}(1990,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) |
| \(\chi_{2601}(2143,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) |
| \(\chi_{2601}(2296,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) |
| \(\chi_{2601}(2449,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) |