from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1000, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,25,9]))
pari: [g,chi] = znchar(Mod(19,1000))
Basic properties
| Modulus: | \(1000\) | |
| Conductor: | \(1000\) | 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 1000.ba
\(\chi_{1000}(19,\cdot)\) \(\chi_{1000}(59,\cdot)\) \(\chi_{1000}(139,\cdot)\) \(\chi_{1000}(179,\cdot)\) \(\chi_{1000}(219,\cdot)\) \(\chi_{1000}(259,\cdot)\) \(\chi_{1000}(339,\cdot)\) \(\chi_{1000}(379,\cdot)\) \(\chi_{1000}(419,\cdot)\) \(\chi_{1000}(459,\cdot)\) \(\chi_{1000}(539,\cdot)\) \(\chi_{1000}(579,\cdot)\) \(\chi_{1000}(619,\cdot)\) \(\chi_{1000}(659,\cdot)\) \(\chi_{1000}(739,\cdot)\) \(\chi_{1000}(779,\cdot)\) \(\chi_{1000}(819,\cdot)\) \(\chi_{1000}(859,\cdot)\) \(\chi_{1000}(939,\cdot)\) \(\chi_{1000}(979,\cdot)\)
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
\((751,501,377)\) → \((-1,-1,e\left(\frac{9}{50}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1000 }(19, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{39}{50}\right)\) |
sage: chi.jacobi_sum(n)