from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8007, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,0,6]))
pari: [g,chi] = znchar(Mod(256,8007))
Basic properties
| Modulus: | \(8007\) | |
| Conductor: | \(157\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(13\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{157}(99,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8007.bw
\(\chi_{8007}(256,\cdot)\) \(\chi_{8007}(1072,\cdot)\) \(\chi_{8007}(1174,\cdot)\) \(\chi_{8007}(1429,\cdot)\) \(\chi_{8007}(1480,\cdot)\) \(\chi_{8007}(2194,\cdot)\) \(\chi_{8007}(2551,\cdot)\) \(\chi_{8007}(4183,\cdot)\) \(\chi_{8007}(4285,\cdot)\) \(\chi_{8007}(4489,\cdot)\) \(\chi_{8007}(5509,\cdot)\) \(\chi_{8007}(5917,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
| Fixed field: | Number field defined by a degree 13 polynomial |
Values on generators
\((5339,1414,7855)\) → \((1,1,e\left(\frac{3}{13}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 8007 }(256, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) |
sage: chi.jacobi_sum(n)