from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6019, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([7,18]))
pari: [g,chi] = znchar(Mod(693,6019))
Basic properties
| Modulus: | \(6019\) | |
| Conductor: | \(6019\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | 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 6019.cc
\(\chi_{6019}(693,\cdot)\) \(\chi_{6019}(771,\cdot)\) \(\chi_{6019}(875,\cdot)\) \(\chi_{6019}(1044,\cdot)\) \(\chi_{6019}(2545,\cdot)\) \(\chi_{6019}(2623,\cdot)\) \(\chi_{6019}(2727,\cdot)\) \(\chi_{6019}(2812,\cdot)\) \(\chi_{6019}(2896,\cdot)\) \(\chi_{6019}(3527,\cdot)\) \(\chi_{6019}(4664,\cdot)\) \(\chi_{6019}(5379,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((2316,1392)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{3}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 6019 }(693, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) |
sage: chi.jacobi_sum(n)