from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8001, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,14,33]))
pari: [g,chi] = znchar(Mod(7604,8001))
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 \(\chi_{2667}(2270,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8001.jb
\(\chi_{8001}(746,\cdot)\) \(\chi_{8001}(1619,\cdot)\) \(\chi_{8001}(2762,\cdot)\) \(\chi_{8001}(5651,\cdot)\) \(\chi_{8001}(5840,\cdot)\) \(\chi_{8001}(6092,\cdot)\) \(\chi_{8001}(6596,\cdot)\) \(\chi_{8001}(6794,\cdot)\) \(\chi_{8001}(6983,\cdot)\) \(\chi_{8001}(7235,\cdot)\) \(\chi_{8001}(7604,\cdot)\) \(\chi_{8001}(7739,\cdot)\)
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
\((3557,1144,7750)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{11}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 8001 }(7604, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)