from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1665, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([2,9,0]))
pari: [g,chi] = znchar(Mod(38,1665))
Basic properties
| Modulus: | \(1665\) | |
| Conductor: | \(45\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{45}(38,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1665.df
\(\chi_{1665}(38,\cdot)\) \(\chi_{1665}(482,\cdot)\) \(\chi_{1665}(1037,\cdot)\) \(\chi_{1665}(1148,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
| Fixed field: | \(\Q(\zeta_{45})^+\) |
Values on generators
\((371,667,631)\) → \((e\left(\frac{1}{6}\right),-i,1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1665 }(38, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(-i\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(i\) | \(-1\) |
sage: chi.jacobi_sum(n)