from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1148, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,10,33]))
pari: [g,chi] = znchar(Mod(87,1148))
Basic properties
| Modulus: | \(1148\) | |
| Conductor: | \(1148\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1148.cg
\(\chi_{1148}(87,\cdot)\) \(\chi_{1148}(103,\cdot)\) \(\chi_{1148}(115,\cdot)\) \(\chi_{1148}(131,\cdot)\) \(\chi_{1148}(143,\cdot)\) \(\chi_{1148}(159,\cdot)\) \(\chi_{1148}(367,\cdot)\) \(\chi_{1148}(535,\cdot)\) \(\chi_{1148}(579,\cdot)\) \(\chi_{1148}(607,\cdot)\) \(\chi_{1148}(635,\cdot)\) \(\chi_{1148}(759,\cdot)\) \(\chi_{1148}(787,\cdot)\) \(\chi_{1148}(815,\cdot)\) \(\chi_{1148}(859,\cdot)\) \(\chi_{1148}(1027,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((575,493,785)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{11}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 1148 }(87, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) |
sage: chi.jacobi_sum(n)