from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6013, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,6]))
pari: [g,chi] = znchar(Mod(463,6013))
Basic properties
| Modulus: | \(6013\) | |
| Conductor: | \(859\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(13\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{859}(463,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6013.v
\(\chi_{6013}(463,\cdot)\) \(\chi_{6013}(1233,\cdot)\) \(\chi_{6013}(1632,\cdot)\) \(\chi_{6013}(1842,\cdot)\) \(\chi_{6013}(2269,\cdot)\) \(\chi_{6013}(3536,\cdot)\) \(\chi_{6013}(3914,\cdot)\) \(\chi_{6013}(3991,\cdot)\) \(\chi_{6013}(4285,\cdot)\) \(\chi_{6013}(5013,\cdot)\) \(\chi_{6013}(5657,\cdot)\) \(\chi_{6013}(5678,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
| Fixed field: | Number field defined by a degree 13 polynomial |
Values on generators
\((5155,3438)\) → \((1,e\left(\frac{3}{13}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 6013 }(463, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) |
sage: chi.jacobi_sum(n)