from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3479, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([29,12]))
pari: [g,chi] = znchar(Mod(250,3479))
Basic properties
Modulus: | \(3479\) | |
Conductor: | \(3479\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3479.cn
\(\chi_{3479}(250,\cdot)\) \(\chi_{3479}(474,\cdot)\) \(\chi_{3479}(1298,\cdot)\) \(\chi_{3479}(1536,\cdot)\) \(\chi_{3479}(1678,\cdot)\) \(\chi_{3479}(1823,\cdot)\) \(\chi_{3479}(1949,\cdot)\) \(\chi_{3479}(2231,\cdot)\) \(\chi_{3479}(2931,\cdot)\) \(\chi_{3479}(3083,\cdot)\) \(\chi_{3479}(3085,\cdot)\) \(\chi_{3479}(3090,\cdot)\)
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
\((640,1569)\) → \((e\left(\frac{29}{42}\right),e\left(\frac{2}{7}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
\( \chi_{ 3479 }(250, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{19}{42}\right)\) |
sage: chi.jacobi_sum(n)