from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1547, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([4,6,3]))
pari: [g,chi] = znchar(Mod(506,1547))
Basic properties
| Modulus: | \(1547\) | |
| Conductor: | \(1547\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1547.co
\(\chi_{1547}(480,\cdot)\) \(\chi_{1547}(506,\cdot)\) \(\chi_{1547}(1143,\cdot)\) \(\chi_{1547}(1390,\cdot)\)
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: | 12.12.3299778027646611785838473.1 |
Values on generators
\((885,834,547)\) → \((e\left(\frac{1}{3}\right),-1,i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 1547 }(506, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(i\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) |
sage: chi.jacobi_sum(n)