from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3312, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,11,12]))
pari: [g,chi] = znchar(Mod(119,3312))
Basic properties
| Modulus: | \(3312\) | |
| Conductor: | \(1656\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1656}(947,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3312.dd
\(\chi_{3312}(119,\cdot)\) \(\chi_{3312}(167,\cdot)\) \(\chi_{3312}(311,\cdot)\) \(\chi_{3312}(407,\cdot)\) \(\chi_{3312}(455,\cdot)\) \(\chi_{3312}(887,\cdot)\) \(\chi_{3312}(1175,\cdot)\) \(\chi_{3312}(1271,\cdot)\) \(\chi_{3312}(1319,\cdot)\) \(\chi_{3312}(1415,\cdot)\) \(\chi_{3312}(1559,\cdot)\) \(\chi_{3312}(1751,\cdot)\) \(\chi_{3312}(1895,\cdot)\) \(\chi_{3312}(1991,\cdot)\) \(\chi_{3312}(2279,\cdot)\) \(\chi_{3312}(2327,\cdot)\) \(\chi_{3312}(2423,\cdot)\) \(\chi_{3312}(2615,\cdot)\) \(\chi_{3312}(2855,\cdot)\) \(\chi_{3312}(2999,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((415,2485,2945,2305)\) → \((-1,-1,e\left(\frac{1}{6}\right),e\left(\frac{2}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3312 }(119, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) |
sage: chi.jacobi_sum(n)