from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1137, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([0,1]))
pari: [g,chi] = znchar(Mod(145,1137))
Basic properties
Modulus: | \(1137\) | |
Conductor: | \(379\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{379}(145,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1137.p
\(\chi_{1137}(40,\cdot)\) \(\chi_{1137}(145,\cdot)\) \(\chi_{1137}(199,\cdot)\) \(\chi_{1137}(295,\cdot)\) \(\chi_{1137}(643,\cdot)\) \(\chi_{1137}(952,\cdot)\)
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
\((380,760)\) → \((1,e\left(\frac{1}{18}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 1137 }(145, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) |
sage: chi.jacobi_sum(n)