from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8004, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,14,14,13]))
pari: [g,chi] = znchar(Mod(275,8004))
Basic properties
| Modulus: | \(8004\) | |
| Conductor: | \(8004\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | 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 8004.cj
\(\chi_{8004}(275,\cdot)\) \(\chi_{8004}(827,\cdot)\) \(\chi_{8004}(1655,\cdot)\) \(\chi_{8004}(2207,\cdot)\) \(\chi_{8004}(2483,\cdot)\) \(\chi_{8004}(3035,\cdot)\) \(\chi_{8004}(4139,\cdot)\) \(\chi_{8004}(4967,\cdot)\) \(\chi_{8004}(6071,\cdot)\) \(\chi_{8004}(6623,\cdot)\) \(\chi_{8004}(6899,\cdot)\) \(\chi_{8004}(7451,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((4003,2669,3133,553)\) → \((-1,-1,-1,e\left(\frac{13}{28}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 8004 }(275, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(-i\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) |
sage: chi.jacobi_sum(n)