from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1600, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([4,7,2]))
pari: [g,chi] = znchar(Mod(407,1600))
Basic properties
Modulus: | \(1600\) | |
Conductor: | \(160\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{160}(147,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1600.v
\(\chi_{1600}(407,\cdot)\) \(\chi_{1600}(743,\cdot)\) \(\chi_{1600}(1207,\cdot)\) \(\chi_{1600}(1543,\cdot)\)
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: | \(\Q(\zeta_{8})\) |
Fixed field: | 8.8.33554432000000.2 |
Values on generators
\((1151,901,577)\) → \((-1,e\left(\frac{7}{8}\right),i)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 1600 }(407, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(-1\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) | \(e\left(\frac{5}{8}\right)\) |
sage: chi.jacobi_sum(n)