from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(529, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([12]))
pari: [g,chi] = znchar(Mod(346,529))
Basic properties
Modulus: | \(529\) | |
Conductor: | \(529\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(23\) | 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 529.e
\(\chi_{529}(24,\cdot)\) \(\chi_{529}(47,\cdot)\) \(\chi_{529}(70,\cdot)\) \(\chi_{529}(93,\cdot)\) \(\chi_{529}(116,\cdot)\) \(\chi_{529}(139,\cdot)\) \(\chi_{529}(162,\cdot)\) \(\chi_{529}(185,\cdot)\) \(\chi_{529}(208,\cdot)\) \(\chi_{529}(231,\cdot)\) \(\chi_{529}(254,\cdot)\) \(\chi_{529}(277,\cdot)\) \(\chi_{529}(300,\cdot)\) \(\chi_{529}(323,\cdot)\) \(\chi_{529}(346,\cdot)\) \(\chi_{529}(369,\cdot)\) \(\chi_{529}(392,\cdot)\) \(\chi_{529}(415,\cdot)\) \(\chi_{529}(438,\cdot)\) \(\chi_{529}(461,\cdot)\) \(\chi_{529}(484,\cdot)\) \(\chi_{529}(507,\cdot)\)
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: | \(\Q(\zeta_{23})\) |
Fixed field: | Number field defined by a degree 23 polynomial |
Values on generators
\(5\) → \(e\left(\frac{6}{23}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 529 }(346, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) |
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)