from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1296, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([0,9,1]))
pari: [g,chi] = znchar(Mod(89,1296))
Basic properties
| Modulus: | \(1296\) | |
| Conductor: | \(216\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{216}(29,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1296.bf
\(\chi_{1296}(89,\cdot)\) \(\chi_{1296}(233,\cdot)\) \(\chi_{1296}(521,\cdot)\) \(\chi_{1296}(665,\cdot)\) \(\chi_{1296}(953,\cdot)\) \(\chi_{1296}(1097,\cdot)\)
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: | 18.0.396521139274783615537700143104.1 |
Values on generators
\((1135,325,1217)\) → \((1,-1,e\left(\frac{1}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1296 }(89, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) |
sage: chi.jacobi_sum(n)