from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1143, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([7,10]))
pari: [g,chi] = znchar(Mod(38,1143))
Basic properties
Modulus: | \(1143\) | |
Conductor: | \(1143\) | 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 1143.br
\(\chi_{1143}(38,\cdot)\) \(\chi_{1143}(122,\cdot)\) \(\chi_{1143}(200,\cdot)\) \(\chi_{1143}(221,\cdot)\) \(\chi_{1143}(227,\cdot)\) \(\chi_{1143}(428,\cdot)\) \(\chi_{1143}(533,\cdot)\) \(\chi_{1143}(752,\cdot)\) \(\chi_{1143}(812,\cdot)\) \(\chi_{1143}(950,\cdot)\) \(\chi_{1143}(965,\cdot)\) \(\chi_{1143}(1103,\cdot)\)
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
\((128,892)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{5}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 1143 }(38, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{21}\right)\) |
sage: chi.jacobi_sum(n)