from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([9,10,0]))
pari: [g,chi] = znchar(Mod(243,8470))
Basic properties
Modulus: | \(8470\) | |
Conductor: | \(35\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{35}(33,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8470.bd
\(\chi_{8470}(243,\cdot)\) \(\chi_{8470}(1937,\cdot)\) \(\chi_{8470}(2663,\cdot)\) \(\chi_{8470}(4357,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
Fixed field: | \(\Q(\zeta_{35})^+\) |
Values on generators
\((6777,6051,7141)\) → \((-i,e\left(\frac{5}{6}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 8470 }(243, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(i\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)