from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3484, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,22,8]))
pari: [g,chi] = znchar(Mod(55,3484))
Basic properties
Modulus: | \(3484\) | |
Conductor: | \(3484\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3484.dg
\(\chi_{3484}(55,\cdot)\) \(\chi_{3484}(315,\cdot)\) \(\chi_{3484}(451,\cdot)\) \(\chi_{3484}(555,\cdot)\) \(\chi_{3484}(1199,\cdot)\) \(\chi_{3484}(1283,\cdot)\) \(\chi_{3484}(1491,\cdot)\) \(\chi_{3484}(1595,\cdot)\) \(\chi_{3484}(1647,\cdot)\) \(\chi_{3484}(1959,\cdot)\) \(\chi_{3484}(1979,\cdot)\) \(\chi_{3484}(2031,\cdot)\) \(\chi_{3484}(2167,\cdot)\) \(\chi_{3484}(2447,\cdot)\) \(\chi_{3484}(3019,\cdot)\) \(\chi_{3484}(3071,\cdot)\) \(\chi_{3484}(3155,\cdot)\) \(\chi_{3484}(3175,\cdot)\) \(\chi_{3484}(3383,\cdot)\) \(\chi_{3484}(3415,\cdot)\)
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
\((1743,1341,3017)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{4}{33}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
\( \chi_{ 3484 }(55, a) \) | \(-1\) | \(1\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) |
sage: chi.jacobi_sum(n)