from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8007, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,3,20]))
pari: [g,chi] = znchar(Mod(145,8007))
Basic properties
Modulus: | \(8007\) | |
Conductor: | \(2669\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2669}(145,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8007.ch
\(\chi_{8007}(145,\cdot)\) \(\chi_{8007}(484,\cdot)\) \(\chi_{8007}(1426,\cdot)\) \(\chi_{8007}(3442,\cdot)\) \(\chi_{8007}(4252,\cdot)\) \(\chi_{8007}(4384,\cdot)\) \(\chi_{8007}(5194,\cdot)\) \(\chi_{8007}(7210,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
Fixed field: | Number field defined by a degree 24 polynomial |
Values on generators
\((5339,1414,7855)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 8007 }(145, a) \) | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)