from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1665, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([6,9,1]))
pari: [g,chi] = znchar(Mod(4,1665))
Basic properties
| Modulus: | \(1665\) | |
| Conductor: | \(1665\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(18\) | 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 1665.eh
\(\chi_{1665}(4,\cdot)\) \(\chi_{1665}(169,\cdot)\) \(\chi_{1665}(1024,\cdot)\) \(\chi_{1665}(1039,\cdot)\) \(\chi_{1665}(1249,\cdot)\) \(\chi_{1665}(1399,\cdot)\)
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: | \(\Q(\zeta_{9})\) |
| Fixed field: | 18.18.251807963938795439839460968737373396634765625.1 |
Values on generators
\((371,667,631)\) → \((e\left(\frac{1}{3}\right),-1,e\left(\frac{1}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1665 }(4, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(-1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) |
sage: chi.jacobi_sum(n)