from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1137, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,16]))
pari: [g,chi] = znchar(Mod(5,1137))
Basic properties
| Modulus: | \(1137\) | |
| Conductor: | \(1137\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1137.t
\(\chi_{1137}(5,\cdot)\) \(\chi_{1137}(311,\cdot)\) \(\chi_{1137}(404,\cdot)\) \(\chi_{1137}(455,\cdot)\) \(\chi_{1137}(470,\cdot)\) \(\chi_{1137}(596,\cdot)\) \(\chi_{1137}(701,\cdot)\) \(\chi_{1137}(797,\cdot)\) \(\chi_{1137}(851,\cdot)\) \(\chi_{1137}(974,\cdot)\) \(\chi_{1137}(1004,\cdot)\) \(\chi_{1137}(1013,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((380,760)\) → \((-1,e\left(\frac{8}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1137 }(5, a) \) | \(-1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{11}{21}\right)\) |
sage: chi.jacobi_sum(n)