from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(507, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,21]))
pari: [g,chi] = znchar(Mod(350,507))
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.n
\(\chi_{507}(38,\cdot)\) \(\chi_{507}(77,\cdot)\) \(\chi_{507}(116,\cdot)\) \(\chi_{507}(155,\cdot)\) \(\chi_{507}(194,\cdot)\) \(\chi_{507}(233,\cdot)\) \(\chi_{507}(272,\cdot)\) \(\chi_{507}(311,\cdot)\) \(\chi_{507}(350,\cdot)\) \(\chi_{507}(389,\cdot)\) \(\chi_{507}(428,\cdot)\) \(\chi_{507}(467,\cdot)\)
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: | 26.0.6106615481290390926196335311405889611927669049273103600389879.1 |
Values on generators
\((170,340)\) → \((-1,e\left(\frac{21}{26}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
\( \chi_{ 507 }(350, a) \) | \(-1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) |
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)