from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4368, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,3,6,0,10]))
pari: [g,chi] = znchar(Mod(491,4368))
Basic properties
| Modulus: | \(4368\) | |
| Conductor: | \(624\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{624}(491,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4368.mu
\(\chi_{4368}(491,\cdot)\) \(\chi_{4368}(1499,\cdot)\) \(\chi_{4368}(2675,\cdot)\) \(\chi_{4368}(3683,\cdot)\)
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: | 12.12.863278466964378177503232.1 |
Values on generators
\((3823,1093,1457,1249,2017)\) → \((-1,i,-1,1,e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4368 }(491, a) \) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(e\left(\frac{7}{12}\right)\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)