from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3312, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,11,11,2]))
pari: [g,chi] = znchar(Mod(71,3312))
Basic properties
| Modulus: | \(3312\) | |
| Conductor: | \(552\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{552}(347,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3312.cd
\(\chi_{3312}(71,\cdot)\) \(\chi_{3312}(215,\cdot)\) \(\chi_{3312}(647,\cdot)\) \(\chi_{3312}(791,\cdot)\) \(\chi_{3312}(1223,\cdot)\) \(\chi_{3312}(1511,\cdot)\) \(\chi_{3312}(2375,\cdot)\) \(\chi_{3312}(2519,\cdot)\) \(\chi_{3312}(2663,\cdot)\) \(\chi_{3312}(3095,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
| Fixed field: | Number field defined by a degree 22 polynomial |
Values on generators
\((415,2485,2945,2305)\) → \((-1,-1,-1,e\left(\frac{1}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3312 }(71, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) |
sage: chi.jacobi_sum(n)