from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3311, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([10,6,5]))
pari: [g,chi] = znchar(Mod(37,3311))
Basic properties
| Modulus: | \(3311\) | |
| Conductor: | \(3311\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | 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)
|
Galois orbit 3311.cz
\(\chi_{3311}(37,\cdot)\) \(\chi_{3311}(179,\cdot)\) \(\chi_{3311}(940,\cdot)\) \(\chi_{3311}(1082,\cdot)\) \(\chi_{3311}(1241,\cdot)\) \(\chi_{3311}(1985,\cdot)\) \(\chi_{3311}(2286,\cdot)\) \(\chi_{3311}(2445,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{15})\) |
| Fixed field: | Number field defined by a degree 30 polynomial |
Values on generators
\((1893,904,2927)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{1}{5}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 3311 }(37, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{8}{15}\right)\) |
sage: chi.jacobi_sum(n)