from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,21,39]))
pari: [g,chi] = znchar(Mod(22,6003))
Basic properties
Modulus: | \(6003\) | |
Conductor: | \(6003\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6003.bu
\(\chi_{6003}(22,\cdot)\) \(\chi_{6003}(643,\cdot)\) \(\chi_{6003}(850,\cdot)\) \(\chi_{6003}(1057,\cdot)\) \(\chi_{6003}(1195,\cdot)\) \(\chi_{6003}(2023,\cdot)\) \(\chi_{6003}(2092,\cdot)\) \(\chi_{6003}(2644,\cdot)\) \(\chi_{6003}(2851,\cdot)\) \(\chi_{6003}(3058,\cdot)\) \(\chi_{6003}(4093,\cdot)\) \(\chi_{6003}(5197,\cdot)\)
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
\((668,3133,4555)\) → \((e\left(\frac{1}{3}\right),-1,e\left(\frac{13}{14}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 6003 }(22, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) |
sage: chi.jacobi_sum(n)