from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2019, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,15]))
pari: [g,chi] = znchar(Mod(56,2019))
Basic properties
Modulus: | \(2019\) | |
Conductor: | \(2019\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | 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 2019.y
\(\chi_{2019}(56,\cdot)\) \(\chi_{2019}(617,\cdot)\) \(\chi_{2019}(710,\cdot)\) \(\chi_{2019}(851,\cdot)\) \(\chi_{2019}(1055,\cdot)\) \(\chi_{2019}(1070,\cdot)\) \(\chi_{2019}(1334,\cdot)\) \(\chi_{2019}(1358,\cdot)\) \(\chi_{2019}(1622,\cdot)\) \(\chi_{2019}(1637,\cdot)\) \(\chi_{2019}(1841,\cdot)\) \(\chi_{2019}(1982,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((674,1351)\) → \((-1,e\left(\frac{15}{28}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 2019 }(56, a) \) | \(-1\) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)