from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3479, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([2,33]))
pari: [g,chi] = znchar(Mod(744,3479))
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.dd
\(\chi_{3479}(744,\cdot)\) \(\chi_{3479}(751,\cdot)\) \(\chi_{3479}(1033,\cdot)\) \(\chi_{3479}(1885,\cdot)\) \(\chi_{3479}(2011,\cdot)\) \(\chi_{3479}(2298,\cdot)\) \(\chi_{3479}(2377,\cdot)\) \(\chi_{3479}(2536,\cdot)\) \(\chi_{3479}(2678,\cdot)\) \(\chi_{3479}(2881,\cdot)\) \(\chi_{3479}(2937,\cdot)\) \(\chi_{3479}(3147,\cdot)\)
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
\((640,1569)\) → \((e\left(\frac{1}{21}\right),e\left(\frac{11}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 3479 }(744, a) \) | \(-1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) |
sage: chi.jacobi_sum(n)