from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3312, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,55,6]))
pari: [g,chi] = znchar(Mod(1175,3312))
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 \(\chi_{1656}(347,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3312.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
\((415,2485,2945,2305)\) → \((-1,-1,e\left(\frac{5}{6}\right),e\left(\frac{1}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 3312 }(1175, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) |
sage: chi.jacobi_sum(n)