from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8007, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,13,16]))
pari: [g,chi] = znchar(Mod(16,8007))
Basic properties
Modulus: | \(8007\) | |
Conductor: | \(2669\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2669}(16,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8007.cp
\(\chi_{8007}(16,\cdot)\) \(\chi_{8007}(67,\cdot)\) \(\chi_{8007}(781,\cdot)\) \(\chi_{8007}(1138,\cdot)\) \(\chi_{8007}(2770,\cdot)\) \(\chi_{8007}(2872,\cdot)\) \(\chi_{8007}(3076,\cdot)\) \(\chi_{8007}(4096,\cdot)\) \(\chi_{8007}(4504,\cdot)\) \(\chi_{8007}(6850,\cdot)\) \(\chi_{8007}(7666,\cdot)\) \(\chi_{8007}(7768,\cdot)\)
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 26 polynomial |
Values on generators
\((5339,1414,7855)\) → \((1,-1,e\left(\frac{8}{13}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 8007 }(16, a) \) | \(1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(1\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) |
sage: chi.jacobi_sum(n)