from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1666, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([8,7]))
pari: [g,chi] = znchar(Mod(183,1666))
Basic properties
Modulus: | \(1666\) | |
Conductor: | \(833\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{833}(183,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1666.x
\(\chi_{1666}(183,\cdot)\) \(\chi_{1666}(225,\cdot)\) \(\chi_{1666}(421,\cdot)\) \(\chi_{1666}(463,\cdot)\) \(\chi_{1666}(659,\cdot)\) \(\chi_{1666}(701,\cdot)\) \(\chi_{1666}(897,\cdot)\) \(\chi_{1666}(939,\cdot)\) \(\chi_{1666}(1135,\cdot)\) \(\chi_{1666}(1415,\cdot)\) \(\chi_{1666}(1611,\cdot)\) \(\chi_{1666}(1653,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((885,785)\) → \((e\left(\frac{2}{7}\right),i)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
\( \chi_{ 1666 }(183, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(-1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) |
sage: chi.jacobi_sum(n)