from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6026, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,2]))
pari: [g,chi] = znchar(Mod(369,6026))
Basic properties
| Modulus: | \(6026\) | |
| Conductor: | \(131\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(13\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{131}(107,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6026.j
\(\chi_{6026}(369,\cdot)\) \(\chi_{6026}(1611,\cdot)\) \(\chi_{6026}(1933,\cdot)\) \(\chi_{6026}(2025,\cdot)\) \(\chi_{6026}(2209,\cdot)\) \(\chi_{6026}(2945,\cdot)\) \(\chi_{6026}(3359,\cdot)\) \(\chi_{6026}(3451,\cdot)\) \(\chi_{6026}(3589,\cdot)\) \(\chi_{6026}(3911,\cdot)\) \(\chi_{6026}(4141,\cdot)\) \(\chi_{6026}(4647,\cdot)\)
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
\((787,4325)\) → \((1,e\left(\frac{1}{13}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 6026 }(369, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) |
sage: chi.jacobi_sum(n)