from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8044, base_ring=CyclotomicField(6))
M = H._module
chi = DirichletCharacter(H, M([0,2]))
pari: [g,chi] = znchar(Mod(205,8044))
Basic properties
Modulus: | \(8044\) | |
Conductor: | \(2011\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(3\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2011}(205,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8044.e
\(\chi_{8044}(205,\cdot)\) \(\chi_{8044}(1805,\cdot)\)
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: | \(\Q(\sqrt{-3}) \) |
Fixed field: | 3.3.4044121.1 |
Values on generators
\((4023,4025)\) → \((1,e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 8044 }(205, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)