from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2015, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,45,32]))
pari: [g,chi] = znchar(Mod(369,2015))
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.gk
\(\chi_{2015}(164,\cdot)\) \(\chi_{2015}(174,\cdot)\) \(\chi_{2015}(359,\cdot)\) \(\chi_{2015}(369,\cdot)\) \(\chi_{2015}(629,\cdot)\) \(\chi_{2015}(824,\cdot)\) \(\chi_{2015}(944,\cdot)\) \(\chi_{2015}(1074,\cdot)\) \(\chi_{2015}(1409,\cdot)\) \(\chi_{2015}(1464,\cdot)\) \(\chi_{2015}(1529,\cdot)\) \(\chi_{2015}(1539,\cdot)\) \(\chi_{2015}(1724,\cdot)\) \(\chi_{2015}(1919,\cdot)\) \(\chi_{2015}(1929,\cdot)\) \(\chi_{2015}(1994,\cdot)\)
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
\((807,1861,716)\) → \((-1,-i,e\left(\frac{8}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 2015 }(369, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) |
sage: chi.jacobi_sum(n)