from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2352, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,21,14,6]))
pari: [g,chi] = znchar(Mod(83,2352))
Basic properties
| Modulus: | \(2352\) | |
| Conductor: | \(2352\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2352.cq
\(\chi_{2352}(83,\cdot)\) \(\chi_{2352}(251,\cdot)\) \(\chi_{2352}(419,\cdot)\) \(\chi_{2352}(755,\cdot)\) \(\chi_{2352}(923,\cdot)\) \(\chi_{2352}(1091,\cdot)\) \(\chi_{2352}(1259,\cdot)\) \(\chi_{2352}(1427,\cdot)\) \(\chi_{2352}(1595,\cdot)\) \(\chi_{2352}(1931,\cdot)\) \(\chi_{2352}(2099,\cdot)\) \(\chi_{2352}(2267,\cdot)\)
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: | 28.0.1299897872366841427508224117886275845758999455352318474379428852957970432.1 |
Values on generators
\((1471,1765,785,2257)\) → \((-1,-i,-1,e\left(\frac{3}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 2352 }(83, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(i\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(1\) | \(e\left(\frac{17}{28}\right)\) |
sage: chi.jacobi_sum(n)