from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1053, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([52,27]))
pari: [g,chi] = znchar(Mod(142,1053))
Basic properties
| Modulus: | \(1053\) | |
| Conductor: | \(1053\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | 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.cd
\(\chi_{1053}(25,\cdot)\) \(\chi_{1053}(103,\cdot)\) \(\chi_{1053}(142,\cdot)\) \(\chi_{1053}(220,\cdot)\) \(\chi_{1053}(259,\cdot)\) \(\chi_{1053}(337,\cdot)\) \(\chi_{1053}(376,\cdot)\) \(\chi_{1053}(454,\cdot)\) \(\chi_{1053}(493,\cdot)\) \(\chi_{1053}(571,\cdot)\) \(\chi_{1053}(610,\cdot)\) \(\chi_{1053}(688,\cdot)\) \(\chi_{1053}(727,\cdot)\) \(\chi_{1053}(805,\cdot)\) \(\chi_{1053}(844,\cdot)\) \(\chi_{1053}(922,\cdot)\) \(\chi_{1053}(961,\cdot)\) \(\chi_{1053}(1039,\cdot)\)
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 54 polynomial |
Values on generators
\((326,730)\) → \((e\left(\frac{26}{27}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 1053 }(142, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) |
sage: chi.jacobi_sum(n)