from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3479, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([20,21]))
pari: [g,chi] = znchar(Mod(212,3479))
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.df
\(\chi_{3479}(212,\cdot)\) \(\chi_{3479}(354,\cdot)\) \(\chi_{3479}(709,\cdot)\) \(\chi_{3479}(1348,\cdot)\) \(\chi_{3479}(1703,\cdot)\) \(\chi_{3479}(1845,\cdot)\) \(\chi_{3479}(2200,\cdot)\) \(\chi_{3479}(2342,\cdot)\) \(\chi_{3479}(2697,\cdot)\) \(\chi_{3479}(2839,\cdot)\) \(\chi_{3479}(3194,\cdot)\) \(\chi_{3479}(3336,\cdot)\)
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
\((640,1569)\) → \((e\left(\frac{10}{21}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 3479 }(212, a) \) | \(-1\) | \(1\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) |
sage: chi.jacobi_sum(n)