from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(205, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([10,39]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,205))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.40.737344664758480175980241689422890987152982395783688920879217945970594882965087890625.2 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{205}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(-i\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) |
| \(\chi_{205}(12,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-i\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) |
| \(\chi_{205}(52,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(-i\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) |
| \(\chi_{205}(58,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(i\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) |
| \(\chi_{205}(63,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(i\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) |
| \(\chi_{205}(88,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(i\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) |
| \(\chi_{205}(108,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(i\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) |
| \(\chi_{205}(112,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(-i\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) |
| \(\chi_{205}(138,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(i\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) |
| \(\chi_{205}(152,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-i\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) |
| \(\chi_{205}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(-i\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) |
| \(\chi_{205}(158,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(i\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) |
| \(\chi_{205}(177,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(-i\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) |
| \(\chi_{205}(183,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(i\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) |
| \(\chi_{205}(188,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(i\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) |
| \(\chi_{205}(192,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(-i\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) |