from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1666, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([8,27]))
pari: [g,chi] = znchar(Mod(31,1666))
Basic properties
| Modulus: | \(1666\) | |
| Conductor: | \(119\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{119}(31,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1666.bc
\(\chi_{1666}(31,\cdot)\) \(\chi_{1666}(129,\cdot)\) \(\chi_{1666}(215,\cdot)\) \(\chi_{1666}(227,\cdot)\) \(\chi_{1666}(313,\cdot)\) \(\chi_{1666}(411,\cdot)\) \(\chi_{1666}(521,\cdot)\) \(\chi_{1666}(607,\cdot)\) \(\chi_{1666}(619,\cdot)\) \(\chi_{1666}(717,\cdot)\) \(\chi_{1666}(913,\cdot)\) \(\chi_{1666}(1195,\cdot)\) \(\chi_{1666}(1391,\cdot)\) \(\chi_{1666}(1489,\cdot)\) \(\chi_{1666}(1501,\cdot)\) \(\chi_{1666}(1587,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((885,785)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{9}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 1666 }(31, a) \) | \(1\) | \(1\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{3}{16}\right)\) |
sage: chi.jacobi_sum(n)