from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(517, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([0,36]))
pari: [g,chi] = znchar(Mod(474,517))
Basic properties
Modulus: | \(517\) | |
Conductor: | \(47\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(23\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{47}(4,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 517.i
\(\chi_{517}(12,\cdot)\) \(\chi_{517}(34,\cdot)\) \(\chi_{517}(56,\cdot)\) \(\chi_{517}(89,\cdot)\) \(\chi_{517}(100,\cdot)\) \(\chi_{517}(111,\cdot)\) \(\chi_{517}(122,\cdot)\) \(\chi_{517}(144,\cdot)\) \(\chi_{517}(155,\cdot)\) \(\chi_{517}(166,\cdot)\) \(\chi_{517}(177,\cdot)\) \(\chi_{517}(243,\cdot)\) \(\chi_{517}(298,\cdot)\) \(\chi_{517}(309,\cdot)\) \(\chi_{517}(331,\cdot)\) \(\chi_{517}(353,\cdot)\) \(\chi_{517}(397,\cdot)\) \(\chi_{517}(408,\cdot)\) \(\chi_{517}(430,\cdot)\) \(\chi_{517}(441,\cdot)\) \(\chi_{517}(474,\cdot)\) \(\chi_{517}(507,\cdot)\)
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
\((189,287)\) → \((1,e\left(\frac{18}{23}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 517 }(474, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) |
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)