from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(159, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,5]))
pari: [g,chi] = znchar(Mod(17,159))
Basic properties
Modulus: | \(159\) | |
Conductor: | \(159\) | 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 159.h
\(\chi_{159}(11,\cdot)\) \(\chi_{159}(17,\cdot)\) \(\chi_{159}(29,\cdot)\) \(\chi_{159}(38,\cdot)\) \(\chi_{159}(59,\cdot)\) \(\chi_{159}(62,\cdot)\) \(\chi_{159}(110,\cdot)\) \(\chi_{159}(113,\cdot)\) \(\chi_{159}(131,\cdot)\) \(\chi_{159}(143,\cdot)\) \(\chi_{159}(146,\cdot)\) \(\chi_{159}(149,\cdot)\)
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: | 26.0.20392621164064374190468609000303545984542460445839.1 |
Values on generators
\((107,55)\) → \((-1,e\left(\frac{5}{26}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 159 }(17, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) |
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)