from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1148, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,20,33]))
pari: [g,chi] = znchar(Mod(415,1148))
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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1148.ce
\(\chi_{1148}(39,\cdot)\) \(\chi_{1148}(207,\cdot)\) \(\chi_{1148}(415,\cdot)\) \(\chi_{1148}(431,\cdot)\) \(\chi_{1148}(443,\cdot)\) \(\chi_{1148}(459,\cdot)\) \(\chi_{1148}(471,\cdot)\) \(\chi_{1148}(487,\cdot)\) \(\chi_{1148}(695,\cdot)\) \(\chi_{1148}(863,\cdot)\) \(\chi_{1148}(907,\cdot)\) \(\chi_{1148}(935,\cdot)\) \(\chi_{1148}(963,\cdot)\) \(\chi_{1148}(1087,\cdot)\) \(\chi_{1148}(1115,\cdot)\) \(\chi_{1148}(1143,\cdot)\)
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
\((575,493,785)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{11}{20}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 1148 }(415, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) |
sage: chi.jacobi_sum(n)