from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2015, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,55,56]))
pari: [g,chi] = znchar(Mod(813,2015))
Basic properties
Modulus: | \(2015\) | |
Conductor: | \(2015\) | 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 2015.hm
\(\chi_{2015}(162,\cdot)\) \(\chi_{2015}(258,\cdot)\) \(\chi_{2015}(293,\cdot)\) \(\chi_{2015}(392,\cdot)\) \(\chi_{2015}(422,\cdot)\) \(\chi_{2015}(617,\cdot)\) \(\chi_{2015}(813,\cdot)\) \(\chi_{2015}(847,\cdot)\) \(\chi_{2015}(908,\cdot)\) \(\chi_{2015}(1073,\cdot)\) \(\chi_{2015}(1103,\cdot)\) \(\chi_{2015}(1268,\cdot)\) \(\chi_{2015}(1497,\cdot)\) \(\chi_{2015}(1657,\cdot)\) \(\chi_{2015}(1692,\cdot)\) \(\chi_{2015}(1818,\cdot)\)
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
\((807,1861,716)\) → \((-i,e\left(\frac{11}{12}\right),e\left(\frac{14}{15}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
\( \chi_{ 2015 }(813, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) |
sage: chi.jacobi_sum(n)