from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1156, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,15]))
pari: [g,chi] = znchar(Mod(747,1156))
Basic properties
Modulus: | \(1156\) | |
Conductor: | \(1156\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(34\) | 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 1156.l
\(\chi_{1156}(67,\cdot)\) \(\chi_{1156}(135,\cdot)\) \(\chi_{1156}(203,\cdot)\) \(\chi_{1156}(271,\cdot)\) \(\chi_{1156}(339,\cdot)\) \(\chi_{1156}(407,\cdot)\) \(\chi_{1156}(475,\cdot)\) \(\chi_{1156}(543,\cdot)\) \(\chi_{1156}(611,\cdot)\) \(\chi_{1156}(679,\cdot)\) \(\chi_{1156}(747,\cdot)\) \(\chi_{1156}(815,\cdot)\) \(\chi_{1156}(883,\cdot)\) \(\chi_{1156}(951,\cdot)\) \(\chi_{1156}(1019,\cdot)\) \(\chi_{1156}(1087,\cdot)\)
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: | \(\Q(\zeta_{17})\) |
Fixed field: | 34.0.1637569504672609029759453328209845791289707218675094773643512138419836077449127814221529088.1 |
Values on generators
\((579,581)\) → \((-1,e\left(\frac{15}{34}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
\( \chi_{ 1156 }(747, a) \) | \(-1\) | \(1\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) |
sage: chi.jacobi_sum(n)