from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8003, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,26]))
pari: [g,chi] = znchar(Mod(5301,8003))
Basic properties
| Modulus: | \(8003\) | |
| Conductor: | \(151\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(15\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{151}(16,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8003.r
\(\chi_{8003}(1061,\cdot)\) \(\chi_{8003}(2068,\cdot)\) \(\chi_{8003}(2492,\cdot)\) \(\chi_{8003}(3022,\cdot)\) \(\chi_{8003}(5301,\cdot)\) \(\chi_{8003}(5672,\cdot)\) \(\chi_{8003}(7527,\cdot)\) \(\chi_{8003}(7739,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | Number field defined by a degree 15 polynomial |
Values on generators
\((4984,7103)\) → \((1,e\left(\frac{13}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8003 }(5301, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) |
sage: chi.jacobi_sum(n)