from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2160, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([9,9,2,9]))
pari: [g,chi] = znchar(Mod(679,2160))
Basic properties
| Modulus: | \(2160\) | |
| Conductor: | \(1080\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1080}(139,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2160.dl
\(\chi_{2160}(439,\cdot)\) \(\chi_{2160}(679,\cdot)\) \(\chi_{2160}(1159,\cdot)\) \(\chi_{2160}(1399,\cdot)\) \(\chi_{2160}(1879,\cdot)\) \(\chi_{2160}(2119,\cdot)\)
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: | Number field defined by a degree 18 polynomial |
Values on generators
\((271,1621,2081,1297)\) → \((-1,-1,e\left(\frac{1}{9}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2160 }(679, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) |
sage: chi.jacobi_sum(n)