from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6336, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,5,20,8]))
pari: [g,chi] = znchar(Mod(1511,6336))
Basic properties
Modulus: | \(6336\) | |
Conductor: | \(1056\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1056}(59,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6336.el
\(\chi_{6336}(71,\cdot)\) \(\chi_{6336}(647,\cdot)\) \(\chi_{6336}(1367,\cdot)\) \(\chi_{6336}(1511,\cdot)\) \(\chi_{6336}(1655,\cdot)\) \(\chi_{6336}(2231,\cdot)\) \(\chi_{6336}(2951,\cdot)\) \(\chi_{6336}(3095,\cdot)\) \(\chi_{6336}(3239,\cdot)\) \(\chi_{6336}(3815,\cdot)\) \(\chi_{6336}(4535,\cdot)\) \(\chi_{6336}(4679,\cdot)\) \(\chi_{6336}(4823,\cdot)\) \(\chi_{6336}(5399,\cdot)\) \(\chi_{6336}(6119,\cdot)\) \(\chi_{6336}(6263,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((4159,4357,3521,1729)\) → \((-1,e\left(\frac{1}{8}\right),-1,e\left(\frac{1}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 6336 }(1511, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(-i\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{23}{40}\right)\) |
sage: chi.jacobi_sum(n)