from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(689, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,24]))
pari: [g,chi] = znchar(Mod(222,689))
Basic properties
Modulus: | \(689\) | |
Conductor: | \(53\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(13\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{53}(10,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 689.u
\(\chi_{689}(66,\cdot)\) \(\chi_{689}(183,\cdot)\) \(\chi_{689}(222,\cdot)\) \(\chi_{689}(248,\cdot)\) \(\chi_{689}(261,\cdot)\) \(\chi_{689}(365,\cdot)\) \(\chi_{689}(417,\cdot)\) \(\chi_{689}(521,\cdot)\) \(\chi_{689}(599,\cdot)\) \(\chi_{689}(625,\cdot)\) \(\chi_{689}(651,\cdot)\) \(\chi_{689}(664,\cdot)\)
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: | Number field defined by a degree 13 polynomial |
Values on generators
\((54,638)\) → \((1,e\left(\frac{12}{13}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 689 }(222, a) \) | \(1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) |
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)