from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8001, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,28,6]))
pari: [g,chi] = znchar(Mod(4,8001))
Basic properties
| Modulus: | \(8001\) | |
| Conductor: | \(8001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | 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)
|
Galois orbit 8001.ho
\(\chi_{8001}(4,\cdot)\) \(\chi_{8001}(16,\cdot)\) \(\chi_{8001}(256,\cdot)\) \(\chi_{8001}(445,\cdot)\) \(\chi_{8001}(1024,\cdot)\) \(\chi_{8001}(1528,\cdot)\) \(\chi_{8001}(1780,\cdot)\) \(\chi_{8001}(1969,\cdot)\) \(\chi_{8001}(4477,\cdot)\) \(\chi_{8001}(6001,\cdot)\) \(\chi_{8001}(6493,\cdot)\) \(\chi_{8001}(7501,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 21 polynomial |
Values on generators
\((3557,1144,7750)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{2}{3}\right),e\left(\frac{1}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 8001 }(4, a) \) | \(1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)