from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3131, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,7]))
pari: [g,chi] = znchar(Mod(123,3131))
Basic properties
| Modulus: | \(3131\) | |
| Conductor: | \(3131\) | 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 3131.de
\(\chi_{3131}(30,\cdot)\) \(\chi_{3131}(123,\cdot)\) \(\chi_{3131}(247,\cdot)\) \(\chi_{3131}(278,\cdot)\) \(\chi_{3131}(526,\cdot)\) \(\chi_{3131}(619,\cdot)\) \(\chi_{3131}(929,\cdot)\) \(\chi_{3131}(991,\cdot)\) \(\chi_{3131}(1053,\cdot)\) \(\chi_{3131}(1115,\cdot)\) \(\chi_{3131}(1611,\cdot)\) \(\chi_{3131}(1766,\cdot)\) \(\chi_{3131}(1952,\cdot)\) \(\chi_{3131}(1983,\cdot)\) \(\chi_{3131}(2231,\cdot)\) \(\chi_{3131}(2572,\cdot)\) \(\chi_{3131}(2696,\cdot)\) \(\chi_{3131}(2851,\cdot)\) \(\chi_{3131}(2913,\cdot)\) \(\chi_{3131}(3006,\cdot)\)
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
\((809,2729)\) → \((-1,e\left(\frac{7}{50}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 3131 }(123, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(-1\) | \(e\left(\frac{8}{25}\right)\) |
sage: chi.jacobi_sum(n)