from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2015, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,55,18]))
pari: [g,chi] = znchar(Mod(618,2015))
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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2015.hl
\(\chi_{2015}(457,\cdot)\) \(\chi_{2015}(488,\cdot)\) \(\chi_{2015}(587,\cdot)\) \(\chi_{2015}(618,\cdot)\) \(\chi_{2015}(1007,\cdot)\) \(\chi_{2015}(1038,\cdot)\) \(\chi_{2015}(1267,\cdot)\) \(\chi_{2015}(1298,\cdot)\) \(\chi_{2015}(1627,\cdot)\) \(\chi_{2015}(1658,\cdot)\) \(\chi_{2015}(1852,\cdot)\) \(\chi_{2015}(1883,\cdot)\) \(\chi_{2015}(1887,\cdot)\) \(\chi_{2015}(1918,\cdot)\) \(\chi_{2015}(1982,\cdot)\) \(\chi_{2015}(2013,\cdot)\)
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
\((807,1861,716)\) → \((-i,e\left(\frac{11}{12}\right),e\left(\frac{3}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
\( \chi_{ 2015 }(618, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) |
sage: chi.jacobi_sum(n)