from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(500, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,6]))
pari: [g,chi] = znchar(Mod(471,500))
Basic properties
Modulus: | \(500\) | |
Conductor: | \(500\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(50\) | 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 500.p
\(\chi_{500}(11,\cdot)\) \(\chi_{500}(31,\cdot)\) \(\chi_{500}(71,\cdot)\) \(\chi_{500}(91,\cdot)\) \(\chi_{500}(111,\cdot)\) \(\chi_{500}(131,\cdot)\) \(\chi_{500}(171,\cdot)\) \(\chi_{500}(191,\cdot)\) \(\chi_{500}(211,\cdot)\) \(\chi_{500}(231,\cdot)\) \(\chi_{500}(271,\cdot)\) \(\chi_{500}(291,\cdot)\) \(\chi_{500}(311,\cdot)\) \(\chi_{500}(331,\cdot)\) \(\chi_{500}(371,\cdot)\) \(\chi_{500}(391,\cdot)\) \(\chi_{500}(411,\cdot)\) \(\chi_{500}(431,\cdot)\) \(\chi_{500}(471,\cdot)\) \(\chi_{500}(491,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
Fixed field: | Number field defined by a degree 50 polynomial |
Values on generators
\((251,377)\) → \((-1,e\left(\frac{3}{25}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 500 }(471, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{1}{50}\right)\) |
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)