from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(500, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,49]))
pari: [g,chi] = znchar(Mod(219,500))
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.n
\(\chi_{500}(19,\cdot)\) \(\chi_{500}(39,\cdot)\) \(\chi_{500}(59,\cdot)\) \(\chi_{500}(79,\cdot)\) \(\chi_{500}(119,\cdot)\) \(\chi_{500}(139,\cdot)\) \(\chi_{500}(159,\cdot)\) \(\chi_{500}(179,\cdot)\) \(\chi_{500}(219,\cdot)\) \(\chi_{500}(239,\cdot)\) \(\chi_{500}(259,\cdot)\) \(\chi_{500}(279,\cdot)\) \(\chi_{500}(319,\cdot)\) \(\chi_{500}(339,\cdot)\) \(\chi_{500}(359,\cdot)\) \(\chi_{500}(379,\cdot)\) \(\chi_{500}(419,\cdot)\) \(\chi_{500}(439,\cdot)\) \(\chi_{500}(459,\cdot)\) \(\chi_{500}(479,\cdot)\)
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
\((251,377)\) → \((-1,e\left(\frac{49}{50}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 500 }(219, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) |
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)