from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4730, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([21,0,2]))
pari: [g,chi] = znchar(Mod(3983,4730))
Basic properties
Modulus: | \(4730\) | |
Conductor: | \(215\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{215}(113,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4730.bu
\(\chi_{4730}(727,\cdot)\) \(\chi_{4730}(1673,\cdot)\) \(\chi_{4730}(1937,\cdot)\) \(\chi_{4730}(2817,\cdot)\) \(\chi_{4730}(2883,\cdot)\) \(\chi_{4730}(3037,\cdot)\) \(\chi_{4730}(3147,\cdot)\) \(\chi_{4730}(3257,\cdot)\) \(\chi_{4730}(3763,\cdot)\) \(\chi_{4730}(3983,\cdot)\) \(\chi_{4730}(4093,\cdot)\) \(\chi_{4730}(4203,\cdot)\)
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: | 28.28.1407829094312471215113334241722636006626629352569580078125.1 |
Values on generators
\((947,431,1981)\) → \((-i,1,e\left(\frac{1}{14}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 4730 }(3983, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(i\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) |
sage: chi.jacobi_sum(n)