from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(53, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([1]))
pari: [g,chi] = znchar(Mod(4,53))
Basic properties
Modulus: | \(53\) | |
Conductor: | \(53\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 53.e
\(\chi_{53}(4,\cdot)\) \(\chi_{53}(6,\cdot)\) \(\chi_{53}(7,\cdot)\) \(\chi_{53}(9,\cdot)\) \(\chi_{53}(11,\cdot)\) \(\chi_{53}(17,\cdot)\) \(\chi_{53}(25,\cdot)\) \(\chi_{53}(29,\cdot)\) \(\chi_{53}(37,\cdot)\) \(\chi_{53}(38,\cdot)\) \(\chi_{53}(40,\cdot)\) \(\chi_{53}(43,\cdot)\)
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 26 polynomial |
Values on generators
\(2\) → \(e\left(\frac{1}{26}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 53 }(4, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) |
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)