from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4256, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([0,0,15,7]))
pari: [g,chi] = znchar(Mod(33,4256))
Basic properties
| Modulus: | \(4256\) | |
| Conductor: | \(133\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{133}(33,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4256.gn
\(\chi_{4256}(33,\cdot)\) \(\chi_{4256}(129,\cdot)\) \(\chi_{4256}(705,\cdot)\) \(\chi_{4256}(1473,\cdot)\) \(\chi_{4256}(2369,\cdot)\) \(\chi_{4256}(3841,\cdot)\)
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: | \(\Q(\zeta_{9})\) |
| Fixed field: | 18.18.26018473705616588670034125883291477.1 |
Values on generators
\((799,2661,3041,3137)\) → \((1,1,e\left(\frac{5}{6}\right),e\left(\frac{7}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 4256 }(33, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)