from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(205, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,9]))
chi.galois_orbit()
[g,chi] = znchar(Mod(19,205))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(205\) | |
| Conductor: | \(205\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.0.75504093671268370020376748996904037084465397328249745498031917667388916015625.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{205}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(-i\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) |
| \(\chi_{205}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(-i\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) |
| \(\chi_{205}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(i\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) |
| \(\chi_{205}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(i\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) |
| \(\chi_{205}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(i\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) |
| \(\chi_{205}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(i\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) |
| \(\chi_{205}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(i\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) |
| \(\chi_{205}(94,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(i\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) |
| \(\chi_{205}(99,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(-i\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) |
| \(\chi_{205}(104,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(-i\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) |
| \(\chi_{205}(129,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(-i\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) |
| \(\chi_{205}(134,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(i\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) |
| \(\chi_{205}(149,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(-i\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) |
| \(\chi_{205}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(-i\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) |
| \(\chi_{205}(194,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(i\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) |
| \(\chi_{205}(199,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(-i\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) |