from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8001, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,28,13]))
pari: [g,chi] = znchar(Mod(305,8001))
Basic properties
| Modulus: | \(8001\) | |
| Conductor: | \(2667\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2667}(305,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8001.jg
\(\chi_{8001}(305,\cdot)\) \(\chi_{8001}(548,\cdot)\) \(\chi_{8001}(1430,\cdot)\) \(\chi_{8001}(1880,\cdot)\) \(\chi_{8001}(3329,\cdot)\) \(\chi_{8001}(3509,\cdot)\) \(\chi_{8001}(3518,\cdot)\) \(\chi_{8001}(4328,\cdot)\) \(\chi_{8001}(4958,\cdot)\) \(\chi_{8001}(5030,\cdot)\) \(\chi_{8001}(5273,\cdot)\) \(\chi_{8001}(7928,\cdot)\)
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 42 polynomial |
Values on generators
\((3557,1144,7750)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{13}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 8001 }(305, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)