from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(471, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,4]))
pari: [g,chi] = znchar(Mod(101,471))
Basic properties
| Modulus: | \(471\) | |
| Conductor: | \(471\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 471.o
\(\chi_{471}(14,\cdot)\) \(\chi_{471}(101,\cdot)\) \(\chi_{471}(173,\cdot)\) \(\chi_{471}(203,\cdot)\) \(\chi_{471}(224,\cdot)\) \(\chi_{471}(287,\cdot)\) \(\chi_{471}(353,\cdot)\) \(\chi_{471}(389,\cdot)\) \(\chi_{471}(407,\cdot)\) \(\chi_{471}(413,\cdot)\) \(\chi_{471}(422,\cdot)\) \(\chi_{471}(467,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
| Fixed field: | 26.0.80198818302747948116281352414518078412749016374883916055923.1 |
Values on generators
\((158,319)\) → \((-1,e\left(\frac{2}{13}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 471 }(101, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(1\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)