from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1053, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([40,18]))
pari: [g,chi] = znchar(Mod(718,1053))
Basic properties
| Modulus: | \(1053\) | |
| Conductor: | \(1053\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(27\) | 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 1053.br
\(\chi_{1053}(16,\cdot)\) \(\chi_{1053}(22,\cdot)\) \(\chi_{1053}(133,\cdot)\) \(\chi_{1053}(139,\cdot)\) \(\chi_{1053}(250,\cdot)\) \(\chi_{1053}(256,\cdot)\) \(\chi_{1053}(367,\cdot)\) \(\chi_{1053}(373,\cdot)\) \(\chi_{1053}(484,\cdot)\) \(\chi_{1053}(490,\cdot)\) \(\chi_{1053}(601,\cdot)\) \(\chi_{1053}(607,\cdot)\) \(\chi_{1053}(718,\cdot)\) \(\chi_{1053}(724,\cdot)\) \(\chi_{1053}(835,\cdot)\) \(\chi_{1053}(841,\cdot)\) \(\chi_{1053}(952,\cdot)\) \(\chi_{1053}(958,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 27 polynomial |
Values on generators
\((326,730)\) → \((e\left(\frac{20}{27}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 1053 }(718, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{1}{9}\right)\) |
sage: chi.jacobi_sum(n)