from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8007, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,3,16]))
pari: [g,chi] = znchar(Mod(326,8007))
Basic properties
| Modulus: | \(8007\) | |
| Conductor: | \(8007\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | 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 8007.da
\(\chi_{8007}(326,\cdot)\) \(\chi_{8007}(929,\cdot)\) \(\chi_{8007}(1268,\cdot)\) \(\chi_{8007}(1400,\cdot)\) \(\chi_{8007}(1739,\cdot)\) \(\chi_{8007}(2681,\cdot)\) \(\chi_{8007}(3152,\cdot)\) \(\chi_{8007}(3284,\cdot)\) \(\chi_{8007}(4094,\cdot)\) \(\chi_{8007}(4226,\cdot)\) \(\chi_{8007}(4697,\cdot)\) \(\chi_{8007}(5639,\cdot)\) \(\chi_{8007}(5978,\cdot)\) \(\chi_{8007}(6110,\cdot)\) \(\chi_{8007}(6449,\cdot)\) \(\chi_{8007}(7052,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((5339,1414,7855)\) → \((-1,e\left(\frac{1}{16}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 8007 }(326, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)