from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1682, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([11]))
pari: [g,chi] = znchar(Mod(63,1682))
Basic properties
| Modulus: | \(1682\) | |
| Conductor: | \(29\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(14\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{29}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1682.e
\(\chi_{1682}(63,\cdot)\) \(\chi_{1682}(267,\cdot)\) \(\chi_{1682}(651,\cdot)\) \(\chi_{1682}(1037,\cdot)\) \(\chi_{1682}(1077,\cdot)\) \(\chi_{1682}(1111,\cdot)\)
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: | \(\Q(\zeta_{7})\) |
| Fixed field: | Number field defined by a degree 14 polynomial |
Values on generators
\(843\) → \(e\left(\frac{11}{14}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1682 }(63, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(-1\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) |
sage: chi.jacobi_sum(n)