from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(205, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([15,8]))
chi.galois_orbit()
[g,chi] = znchar(Mod(18,205))
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: | \(20\) | 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: | \(\Q(\zeta_{20})\) |
Fixed field: | 20.0.1945771207112214793287128936767578125.1 |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{205}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(i\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(-1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) |
\(\chi_{205}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(-i\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(-1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) |
\(\chi_{205}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(-i\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(-1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) |
\(\chi_{205}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(i\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(-1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) |
\(\chi_{205}(92,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(-i\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(-1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) |
\(\chi_{205}(98,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(i\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(-1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) |
\(\chi_{205}(133,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(i\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(-1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) |
\(\chi_{205}(182,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(-i\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(-1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) |