from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3104, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,24,25]))
pari: [g,chi] = znchar(Mod(751,3104))
Basic properties
| Modulus: | \(3104\) | |
| Conductor: | \(776\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{776}(363,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3104.et
\(\chi_{3104}(335,\cdot)\) \(\chi_{3104}(399,\cdot)\) \(\chi_{3104}(751,\cdot)\) \(\chi_{3104}(1167,\cdot)\) \(\chi_{3104}(1263,\cdot)\) \(\chi_{3104}(1327,\cdot)\) \(\chi_{3104}(1423,\cdot)\) \(\chi_{3104}(1487,\cdot)\) \(\chi_{3104}(1583,\cdot)\) \(\chi_{3104}(1647,\cdot)\) \(\chi_{3104}(1743,\cdot)\) \(\chi_{3104}(2159,\cdot)\) \(\chi_{3104}(2511,\cdot)\) \(\chi_{3104}(2575,\cdot)\) \(\chi_{3104}(2959,\cdot)\) \(\chi_{3104}(3055,\cdot)\)
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
\((2911,389,2721)\) → \((-1,-1,e\left(\frac{25}{48}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 3104 }(751, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{48}\right)\) |
sage: chi.jacobi_sum(n)