from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1000, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,0,23]))
pari: [g,chi] = znchar(Mod(39,1000))
Basic properties
| Modulus: | \(1000\) | |
| Conductor: | \(500\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{500}(39,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1000.z
\(\chi_{1000}(39,\cdot)\) \(\chi_{1000}(79,\cdot)\) \(\chi_{1000}(119,\cdot)\) \(\chi_{1000}(159,\cdot)\) \(\chi_{1000}(239,\cdot)\) \(\chi_{1000}(279,\cdot)\) \(\chi_{1000}(319,\cdot)\) \(\chi_{1000}(359,\cdot)\) \(\chi_{1000}(439,\cdot)\) \(\chi_{1000}(479,\cdot)\) \(\chi_{1000}(519,\cdot)\) \(\chi_{1000}(559,\cdot)\) \(\chi_{1000}(639,\cdot)\) \(\chi_{1000}(679,\cdot)\) \(\chi_{1000}(719,\cdot)\) \(\chi_{1000}(759,\cdot)\) \(\chi_{1000}(839,\cdot)\) \(\chi_{1000}(879,\cdot)\) \(\chi_{1000}(919,\cdot)\) \(\chi_{1000}(959,\cdot)\)
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
\((751,501,377)\) → \((-1,1,e\left(\frac{23}{50}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1000 }(39, a) \) | \(-1\) | \(1\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) |
sage: chi.jacobi_sum(n)