from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(507, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,16]))
pari: [g,chi] = znchar(Mod(482,507))
Basic properties
Modulus: | \(507\) | |
Conductor: | \(507\) | 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 507.o
\(\chi_{507}(14,\cdot)\) \(\chi_{507}(53,\cdot)\) \(\chi_{507}(92,\cdot)\) \(\chi_{507}(131,\cdot)\) \(\chi_{507}(209,\cdot)\) \(\chi_{507}(248,\cdot)\) \(\chi_{507}(287,\cdot)\) \(\chi_{507}(326,\cdot)\) \(\chi_{507}(365,\cdot)\) \(\chi_{507}(404,\cdot)\) \(\chi_{507}(443,\cdot)\) \(\chi_{507}(482,\cdot)\)
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: | 26.0.469739652406953148168948870108145354763666849944084892337683.1 |
Values on generators
\((170,340)\) → \((-1,e\left(\frac{8}{13}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
\( \chi_{ 507 }(482, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) |
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)