from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6013, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,19]))
pari: [g,chi] = znchar(Mod(313,6013))
Basic properties
| Modulus: | \(6013\) | |
| Conductor: | \(6013\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | 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 6013.bu
\(\chi_{6013}(313,\cdot)\) \(\chi_{6013}(817,\cdot)\) \(\chi_{6013}(878,\cdot)\) \(\chi_{6013}(915,\cdot)\) \(\chi_{6013}(1587,\cdot)\) \(\chi_{6013}(2229,\cdot)\) \(\chi_{6013}(2348,\cdot)\) \(\chi_{6013}(2824,\cdot)\) \(\chi_{6013}(3041,\cdot)\) \(\chi_{6013}(3631,\cdot)\) \(\chi_{6013}(3895,\cdot)\) \(\chi_{6013}(3897,\cdot)\) \(\chi_{6013}(4070,\cdot)\) \(\chi_{6013}(4252,\cdot)\) \(\chi_{6013}(4275,\cdot)\) \(\chi_{6013}(4310,\cdot)\) \(\chi_{6013}(4595,\cdot)\) \(\chi_{6013}(4793,\cdot)\) \(\chi_{6013}(5787,\cdot)\) \(\chi_{6013}(5967,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((5155,3438)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{19}{66}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 6013 }(313, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)