from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3696, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,3,6,8,0]))
pari: [g,chi] = znchar(Mod(683,3696))
Basic properties
| Modulus: | \(3696\) | |
| Conductor: | \(336\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{336}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3696.ew
\(\chi_{3696}(683,\cdot)\) \(\chi_{3696}(947,\cdot)\) \(\chi_{3696}(2531,\cdot)\) \(\chi_{3696}(2795,\cdot)\)
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: | 12.12.36099543110378323968.1 |
Values on generators
\((463,2773,2465,1585,673)\) → \((-1,i,-1,e\left(\frac{2}{3}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3696 }(683, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(-i\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(i\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)