from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12352, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,3,14]))
pari: [g,chi] = znchar(Mod(10045,12352))
Basic properties
| Modulus: | \(12352\) | |
| Conductor: | \(12352\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | 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 12352.dr
\(\chi_{12352}(1973,\cdot)\) \(\chi_{12352}(3045,\cdot)\) \(\chi_{12352}(3869,\cdot)\) \(\chi_{12352}(4237,\cdot)\) \(\chi_{12352}(8149,\cdot)\) \(\chi_{12352}(9221,\cdot)\) \(\chi_{12352}(10045,\cdot)\) \(\chi_{12352}(10413,\cdot)\)
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: | \(\Q(\zeta_{16})\) |
| Fixed field: | Number field defined by a degree 16 polynomial |
Values on generators
\((11967,773,11585)\) → \((1,e\left(\frac{3}{16}\right),e\left(\frac{7}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 12352 }(10045, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) |
sage: chi.jacobi_sum(n)