from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4730, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([21,14,24]))
pari: [g,chi] = znchar(Mod(1913,4730))
Basic properties
Modulus: | \(4730\) | |
Conductor: | \(2365\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2365}(1913,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4730.bv
\(\chi_{4730}(527,\cdot)\) \(\chi_{4730}(637,\cdot)\) \(\chi_{4730}(747,\cdot)\) \(\chi_{4730}(967,\cdot)\) \(\chi_{4730}(1473,\cdot)\) \(\chi_{4730}(1583,\cdot)\) \(\chi_{4730}(1693,\cdot)\) \(\chi_{4730}(1847,\cdot)\) \(\chi_{4730}(1913,\cdot)\) \(\chi_{4730}(2793,\cdot)\) \(\chi_{4730}(3057,\cdot)\) \(\chi_{4730}(4003,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((947,431,1981)\) → \((-i,-1,e\left(\frac{6}{7}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 4730 }(1913, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(i\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) |
sage: chi.jacobi_sum(n)