from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6019, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([7,16]))
pari: [g,chi] = znchar(Mod(641,6019))
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.ca
\(\chi_{6019}(641,\cdot)\) \(\chi_{6019}(1089,\cdot)\) \(\chi_{6019}(1122,\cdot)\) \(\chi_{6019}(1375,\cdot)\) \(\chi_{6019}(2103,\cdot)\) \(\chi_{6019}(2968,\cdot)\) \(\chi_{6019}(3410,\cdot)\) \(\chi_{6019}(3423,\cdot)\) \(\chi_{6019}(3559,\cdot)\) \(\chi_{6019}(5080,\cdot)\) \(\chi_{6019}(5126,\cdot)\) \(\chi_{6019}(5756,\cdot)\)
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
\((2316,1392)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{8}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 6019 }(641, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) |
sage: chi.jacobi_sum(n)