from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1785, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,6,8,21]))
pari: [g,chi] = znchar(Mod(2,1785))
Basic properties
| Modulus: | \(1785\) | |
| Conductor: | \(1785\) | 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 1785.ei
\(\chi_{1785}(2,\cdot)\) \(\chi_{1785}(32,\cdot)\) \(\chi_{1785}(128,\cdot)\) \(\chi_{1785}(263,\cdot)\) \(\chi_{1785}(767,\cdot)\) \(\chi_{1785}(893,\cdot)\) \(\chi_{1785}(1052,\cdot)\) \(\chi_{1785}(1283,\cdot)\)
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
\((596,1072,766,1261)\) → \((-1,i,e\left(\frac{1}{3}\right),e\left(\frac{7}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(19\) | \(22\) | \(23\) | \(26\) |
| \( \chi_{ 1785 }(2, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{23}{24}\right)\) | \(i\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) |
sage: chi.jacobi_sum(n)