from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3660, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,0,15,9]))
chi.galois_orbit()
[g,chi] = znchar(Mod(343,3660))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3660\) | |
| Conductor: | \(1220\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1220.cg | 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: | 20.0.266941959144749707507722683125783712000000000000000.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3660}(343,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(i\) | \(i\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(i\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) |
| \(\chi_{3660}(403,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(i\) | \(i\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(i\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) |
| \(\chi_{3660}(907,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(-i\) | \(-i\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(-i\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) |
| \(\chi_{3660}(943,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(i\) | \(i\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(i\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) |
| \(\chi_{3660}(2107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(-i\) | \(-i\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(-i\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) |
| \(\chi_{3660}(2143,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(i\) | \(i\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(i\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) |
| \(\chi_{3660}(2647,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(-i\) | \(-i\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(-i\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) |
| \(\chi_{3660}(2707,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(-i\) | \(-i\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-i\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) |