from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4851, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([28,5,21]))
pari: [g,chi] = znchar(Mod(439,4851))
Basic properties
Modulus: | \(4851\) | |
Conductor: | \(4851\) | 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 4851.ed
\(\chi_{4851}(439,\cdot)\) \(\chi_{4851}(1132,\cdot)\) \(\chi_{4851}(1165,\cdot)\) \(\chi_{4851}(1825,\cdot)\) \(\chi_{4851}(1858,\cdot)\) \(\chi_{4851}(2551,\cdot)\) \(\chi_{4851}(3211,\cdot)\) \(\chi_{4851}(3244,\cdot)\) \(\chi_{4851}(3904,\cdot)\) \(\chi_{4851}(3937,\cdot)\) \(\chi_{4851}(4597,\cdot)\) \(\chi_{4851}(4630,\cdot)\)
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
\((4313,199,442)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{5}{42}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 4851 }(439, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{42}\right)\) |
sage: chi.jacobi_sum(n)