from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6017, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,6]))
pari: [g,chi] = znchar(Mod(353,6017))
Basic properties
Modulus: | \(6017\) | |
Conductor: | \(547\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(13\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{547}(353,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6017.n
\(\chi_{6017}(353,\cdot)\) \(\chi_{6017}(375,\cdot)\) \(\chi_{6017}(1387,\cdot)\) \(\chi_{6017}(2234,\cdot)\) \(\chi_{6017}(2663,\cdot)\) \(\chi_{6017}(2707,\cdot)\) \(\chi_{6017}(3543,\cdot)\) \(\chi_{6017}(4269,\cdot)\) \(\chi_{6017}(4346,\cdot)\) \(\chi_{6017}(4885,\cdot)\) \(\chi_{6017}(5160,\cdot)\) \(\chi_{6017}(5820,\cdot)\)
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
\((3830,2190)\) → \((1,e\left(\frac{3}{13}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 6017 }(353, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) |
sage: chi.jacobi_sum(n)