from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8003, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,46]))
pari: [g,chi] = znchar(Mod(23,8003))
Basic properties
| Modulus: | \(8003\) | |
| Conductor: | \(8003\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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 8003.bi
\(\chi_{8003}(23,\cdot)\) \(\chi_{8003}(348,\cdot)\) \(\chi_{8003}(1355,\cdot)\) \(\chi_{8003}(1878,\cdot)\) \(\chi_{8003}(2249,\cdot)\) \(\chi_{8003}(2680,\cdot)\) \(\chi_{8003}(2892,\cdot)\) \(\chi_{8003}(4528,\cdot)\) \(\chi_{8003}(4747,\cdot)\) \(\chi_{8003}(5058,\cdot)\) \(\chi_{8003}(5118,\cdot)\) \(\chi_{8003}(5482,\cdot)\) \(\chi_{8003}(6489,\cdot)\) \(\chi_{8003}(7397,\cdot)\) \(\chi_{8003}(7814,\cdot)\) \(\chi_{8003}(7927,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((4984,7103)\) → \((-i,e\left(\frac{23}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8003 }(23, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(i\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) |
sage: chi.jacobi_sum(n)