from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2015, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,5,44]))
pari: [g,chi] = znchar(Mod(1471,2015))
Basic properties
| Modulus: | \(2015\) | |
| Conductor: | \(403\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{403}(262,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2015.hd
\(\chi_{2015}(71,\cdot)\) \(\chi_{2015}(76,\cdot)\) \(\chi_{2015}(111,\cdot)\) \(\chi_{2015}(236,\cdot)\) \(\chi_{2015}(266,\cdot)\) \(\chi_{2015}(431,\cdot)\) \(\chi_{2015}(596,\cdot)\) \(\chi_{2015}(826,\cdot)\) \(\chi_{2015}(856,\cdot)\) \(\chi_{2015}(981,\cdot)\) \(\chi_{2015}(1051,\cdot)\) \(\chi_{2015}(1281,\cdot)\) \(\chi_{2015}(1436,\cdot)\) \(\chi_{2015}(1471,\cdot)\) \(\chi_{2015}(1931,\cdot)\) \(\chi_{2015}(1991,\cdot)\)
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
\((807,1861,716)\) → \((1,e\left(\frac{1}{12}\right),e\left(\frac{11}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 2015 }(1471, a) \) | \(-1\) | \(1\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(-i\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) |
sage: chi.jacobi_sum(n)