from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1856, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,21,11]))
pari: [g,chi] = znchar(Mod(47,1856))
Basic properties
| Modulus: | \(1856\) | |
| Conductor: | \(464\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{464}(163,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1856.bs
\(\chi_{1856}(47,\cdot)\) \(\chi_{1856}(79,\cdot)\) \(\chi_{1856}(559,\cdot)\) \(\chi_{1856}(591,\cdot)\) \(\chi_{1856}(751,\cdot)\) \(\chi_{1856}(943,\cdot)\) \(\chi_{1856}(1071,\cdot)\) \(\chi_{1856}(1199,\cdot)\) \(\chi_{1856}(1295,\cdot)\) \(\chi_{1856}(1423,\cdot)\) \(\chi_{1856}(1551,\cdot)\) \(\chi_{1856}(1743,\cdot)\)
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: | 28.28.461376647922644386311538110285539877693458386764746599325237248.2 |
Values on generators
\((639,581,321)\) → \((-1,-i,e\left(\frac{11}{28}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1856 }(47, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(i\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) |
sage: chi.jacobi_sum(n)