from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4013, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([19]))
pari: [g,chi] = znchar(Mod(10,4013))
Basic properties
Modulus: | \(4013\) | |
Conductor: | \(4013\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(34\) | 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 4013.e
\(\chi_{4013}(10,\cdot)\) \(\chi_{4013}(530,\cdot)\) \(\chi_{4013}(831,\cdot)\) \(\chi_{4013}(1000,\cdot)\) \(\chi_{4013}(1204,\cdot)\) \(\chi_{4013}(2039,\cdot)\) \(\chi_{4013}(2726,\cdot)\) \(\chi_{4013}(2840,\cdot)\) \(\chi_{4013}(3090,\cdot)\) \(\chi_{4013}(3250,\cdot)\) \(\chi_{4013}(3617,\cdot)\) \(\chi_{4013}(3688,\cdot)\) \(\chi_{4013}(3704,\cdot)\) \(\chi_{4013}(3729,\cdot)\) \(\chi_{4013}(3913,\cdot)\) \(\chi_{4013}(3960,\cdot)\)
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: | \(\Q(\zeta_{17})\) |
Fixed field: | Number field defined by a degree 34 polynomial |
Values on generators
\(2\) → \(e\left(\frac{19}{34}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 4013 }(10, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) |
sage: chi.jacobi_sum(n)