from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1156, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,26]))
pari: [g,chi] = znchar(Mod(1055,1156))
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.m
\(\chi_{1156}(35,\cdot)\) \(\chi_{1156}(103,\cdot)\) \(\chi_{1156}(171,\cdot)\) \(\chi_{1156}(239,\cdot)\) \(\chi_{1156}(307,\cdot)\) \(\chi_{1156}(375,\cdot)\) \(\chi_{1156}(443,\cdot)\) \(\chi_{1156}(511,\cdot)\) \(\chi_{1156}(647,\cdot)\) \(\chi_{1156}(715,\cdot)\) \(\chi_{1156}(783,\cdot)\) \(\chi_{1156}(851,\cdot)\) \(\chi_{1156}(919,\cdot)\) \(\chi_{1156}(987,\cdot)\) \(\chi_{1156}(1055,\cdot)\) \(\chi_{1156}(1123,\cdot)\)
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: | 34.0.96327617921918178221144313424108575958218071686770280802559537554108004555831047895384064.1 |
Values on generators
\((579,581)\) → \((-1,e\left(\frac{13}{17}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 1156 }(1055, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) |
sage: chi.jacobi_sum(n)