from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6815, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,1,0]))
pari: [g,chi] = znchar(Mod(2351,6815))
Basic properties
| Modulus: | \(6815\) | |
| Conductor: | \(29\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{29}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6815.bk
\(\chi_{6815}(706,\cdot)\) \(\chi_{6815}(1411,\cdot)\) \(\chi_{6815}(2351,\cdot)\) \(\chi_{6815}(2821,\cdot)\) \(\chi_{6815}(3056,\cdot)\) \(\chi_{6815}(3291,\cdot)\) \(\chi_{6815}(4231,\cdot)\) \(\chi_{6815}(4701,\cdot)\) \(\chi_{6815}(5641,\cdot)\) \(\chi_{6815}(5876,\cdot)\) \(\chi_{6815}(6111,\cdot)\) \(\chi_{6815}(6581,\cdot)\)
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
\((2727,2351,146)\) → \((1,e\left(\frac{1}{28}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 6815 }(2351, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(i\) | \(e\left(\frac{9}{14}\right)\) |
sage: chi.jacobi_sum(n)