from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6017, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,25]))
pari: [g,chi] = znchar(Mod(1748,6017))
Basic properties
| Modulus: | \(6017\) | |
| Conductor: | \(6017\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | 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 6017.t
\(\chi_{6017}(197,\cdot)\) \(\chi_{6017}(857,\cdot)\) \(\chi_{6017}(1132,\cdot)\) \(\chi_{6017}(1671,\cdot)\) \(\chi_{6017}(1748,\cdot)\) \(\chi_{6017}(2474,\cdot)\) \(\chi_{6017}(3310,\cdot)\) \(\chi_{6017}(3354,\cdot)\) \(\chi_{6017}(3783,\cdot)\) \(\chi_{6017}(4630,\cdot)\) \(\chi_{6017}(5642,\cdot)\) \(\chi_{6017}(5664,\cdot)\)
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 26 polynomial |
Values on generators
\((3830,2190)\) → \((-1,e\left(\frac{25}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 6017 }(1748, a) \) | \(1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(-1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(1\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) |
sage: chi.jacobi_sum(n)