from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(517, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,2]))
pari: [g,chi] = znchar(Mod(307,517))
Basic properties
Modulus: | \(517\) | |
Conductor: | \(517\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(46\) | 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 517.k
\(\chi_{517}(21,\cdot)\) \(\chi_{517}(32,\cdot)\) \(\chi_{517}(54,\cdot)\) \(\chi_{517}(65,\cdot)\) \(\chi_{517}(98,\cdot)\) \(\chi_{517}(131,\cdot)\) \(\chi_{517}(153,\cdot)\) \(\chi_{517}(175,\cdot)\) \(\chi_{517}(197,\cdot)\) \(\chi_{517}(230,\cdot)\) \(\chi_{517}(241,\cdot)\) \(\chi_{517}(252,\cdot)\) \(\chi_{517}(263,\cdot)\) \(\chi_{517}(285,\cdot)\) \(\chi_{517}(296,\cdot)\) \(\chi_{517}(307,\cdot)\) \(\chi_{517}(318,\cdot)\) \(\chi_{517}(384,\cdot)\) \(\chi_{517}(439,\cdot)\) \(\chi_{517}(450,\cdot)\) \(\chi_{517}(472,\cdot)\) \(\chi_{517}(494,\cdot)\)
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 46 polynomial |
Values on generators
\((189,287)\) → \((-1,e\left(\frac{1}{23}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 517 }(307, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{10}{23}\right)\) |
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)