from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(500, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,1]))
pari: [g,chi] = znchar(Mod(129,500))
Basic properties
| Modulus: | \(500\) | |
| Conductor: | \(125\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{125}(4,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 500.o
\(\chi_{500}(9,\cdot)\) \(\chi_{500}(29,\cdot)\) \(\chi_{500}(69,\cdot)\) \(\chi_{500}(89,\cdot)\) \(\chi_{500}(109,\cdot)\) \(\chi_{500}(129,\cdot)\) \(\chi_{500}(169,\cdot)\) \(\chi_{500}(189,\cdot)\) \(\chi_{500}(209,\cdot)\) \(\chi_{500}(229,\cdot)\) \(\chi_{500}(269,\cdot)\) \(\chi_{500}(289,\cdot)\) \(\chi_{500}(309,\cdot)\) \(\chi_{500}(329,\cdot)\) \(\chi_{500}(369,\cdot)\) \(\chi_{500}(389,\cdot)\) \(\chi_{500}(409,\cdot)\) \(\chi_{500}(429,\cdot)\) \(\chi_{500}(469,\cdot)\) \(\chi_{500}(489,\cdot)\)
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
\((251,377)\) → \((1,e\left(\frac{1}{50}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 500 }(129, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{21}{50}\right)\) |
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)