from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3332, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,17,21]))
pari: [g,chi] = znchar(Mod(271,3332))
Basic properties
Modulus: | \(3332\) | |
Conductor: | \(3332\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3332.ca
\(\chi_{3332}(271,\cdot)\) \(\chi_{3332}(339,\cdot)\) \(\chi_{3332}(747,\cdot)\) \(\chi_{3332}(1223,\cdot)\) \(\chi_{3332}(1291,\cdot)\) \(\chi_{3332}(1699,\cdot)\) \(\chi_{3332}(1767,\cdot)\) \(\chi_{3332}(2243,\cdot)\) \(\chi_{3332}(2651,\cdot)\) \(\chi_{3332}(2719,\cdot)\) \(\chi_{3332}(3127,\cdot)\) \(\chi_{3332}(3195,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((1667,885,785)\) → \((-1,e\left(\frac{17}{42}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
\( \chi_{ 3332 }(271, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) |
sage: chi.jacobi_sum(n)