from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(119952, base_ring=CyclotomicField(6))
M = H._module
chi = DirichletCharacter(H, M([0,0,1,0,0]))
pari: [g,chi] = znchar(Mod(106625,119952))
Basic properties
Modulus: | \(119952\) | |
Conductor: | \(9\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(6\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{9}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 119952.gx
\(\chi_{119952}(66641,\cdot)\) \(\chi_{119952}(106625,\cdot)\)
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: | \(\Q(\zeta_{9})\) |
Values on generators
\((104959,29989,106625,117505,14113)\) → \((1,1,e\left(\frac{1}{6}\right),1,1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 119952 }(106625, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)