from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4851, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,26,0]))
pari: [g,chi] = znchar(Mod(100,4851))
Basic properties
| Modulus: | \(4851\) | |
| Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{49}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4851.cl
\(\chi_{4851}(100,\cdot)\) \(\chi_{4851}(298,\cdot)\) \(\chi_{4851}(793,\cdot)\) \(\chi_{4851}(991,\cdot)\) \(\chi_{4851}(1486,\cdot)\) \(\chi_{4851}(2179,\cdot)\) \(\chi_{4851}(2377,\cdot)\) \(\chi_{4851}(3070,\cdot)\) \(\chi_{4851}(3565,\cdot)\) \(\chi_{4851}(3763,\cdot)\) \(\chi_{4851}(4258,\cdot)\) \(\chi_{4851}(4456,\cdot)\)
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
\((4313,199,442)\) → \((1,e\left(\frac{13}{21}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 4851 }(100, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{7}\right)\) |
sage: chi.jacobi_sum(n)