from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1120, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,21,18,8]))
pari: [g,chi] = znchar(Mod(723,1120))
Basic properties
| Modulus: | \(1120\) | |
| Conductor: | \(1120\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1120.dy
\(\chi_{1120}(107,\cdot)\) \(\chi_{1120}(163,\cdot)\) \(\chi_{1120}(347,\cdot)\) \(\chi_{1120}(403,\cdot)\) \(\chi_{1120}(667,\cdot)\) \(\chi_{1120}(723,\cdot)\) \(\chi_{1120}(907,\cdot)\) \(\chi_{1120}(963,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | Number field defined by a degree 24 polynomial |
Values on generators
\((351,421,897,801)\) → \((-1,e\left(\frac{7}{8}\right),-i,e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 1120 }(723, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)