from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4851, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([28,32,0]))
pari: [g,chi] = znchar(Mod(331,4851))
Basic properties
Modulus: | \(4851\) | |
Conductor: | \(441\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{441}(331,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4851.cj
\(\chi_{4851}(331,\cdot)\) \(\chi_{4851}(760,\cdot)\) \(\chi_{4851}(1024,\cdot)\) \(\chi_{4851}(1453,\cdot)\) \(\chi_{4851}(1717,\cdot)\) \(\chi_{4851}(2146,\cdot)\) \(\chi_{4851}(2410,\cdot)\) \(\chi_{4851}(2839,\cdot)\) \(\chi_{4851}(3103,\cdot)\) \(\chi_{4851}(3532,\cdot)\) \(\chi_{4851}(3796,\cdot)\) \(\chi_{4851}(4225,\cdot)\)
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 21 polynomial |
Values on generators
\((4313,199,442)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{16}{21}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 4851 }(331, a) \) | \(1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{21}\right)\) |
sage: chi.jacobi_sum(n)