from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1568, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,9,16]))
pari: [g,chi] = znchar(Mod(67,1568))
Basic properties
| Modulus: | \(1568\) | |
| Conductor: | \(224\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{224}(67,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1568.bo
\(\chi_{1568}(67,\cdot)\) \(\chi_{1568}(275,\cdot)\) \(\chi_{1568}(459,\cdot)\) \(\chi_{1568}(667,\cdot)\) \(\chi_{1568}(851,\cdot)\) \(\chi_{1568}(1059,\cdot)\) \(\chi_{1568}(1243,\cdot)\) \(\chi_{1568}(1451,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | 24.0.329123002999201416128761938882499016916992.1 |
Values on generators
\((1471,197,1473)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 1568 }(67, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)