from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3104, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,42,23]))
pari: [g,chi] = znchar(Mod(1971,3104))
Basic properties
| Modulus: | \(3104\) | |
| Conductor: | \(3104\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | 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 3104.er
\(\chi_{3104}(3,\cdot)\) \(\chi_{3104}(219,\cdot)\) \(\chi_{3104}(243,\cdot)\) \(\chi_{3104}(339,\cdot)\) \(\chi_{3104}(579,\cdot)\) \(\chi_{3104}(1035,\cdot)\) \(\chi_{3104}(1099,\cdot)\) \(\chi_{3104}(1347,\cdot)\) \(\chi_{3104}(1715,\cdot)\) \(\chi_{3104}(1915,\cdot)\) \(\chi_{3104}(1971,\cdot)\) \(\chi_{3104}(2187,\cdot)\) \(\chi_{3104}(2339,\cdot)\) \(\chi_{3104}(2427,\cdot)\) \(\chi_{3104}(2811,\cdot)\) \(\chi_{3104}(3051,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((2911,389,2721)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{23}{48}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 3104 }(1971, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{37}{48}\right)\) |
sage: chi.jacobi_sum(n)