from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3312, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([0,3,8,6]))
pari: [g,chi] = znchar(Mod(2437,3312))
Basic properties
Modulus: | \(3312\) | |
Conductor: | \(3312\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3312.bp
\(\chi_{3312}(229,\cdot)\) \(\chi_{3312}(781,\cdot)\) \(\chi_{3312}(1885,\cdot)\) \(\chi_{3312}(2437,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
Fixed field: | 12.0.54739011255265747059867648.1 |
Values on generators
\((415,2485,2945,2305)\) → \((1,i,e\left(\frac{2}{3}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 3312 }(2437, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(-1\) | \(i\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)