from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1053, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([2,9]))
pari: [g,chi] = znchar(Mod(4,1053))
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.ce
\(\chi_{1053}(4,\cdot)\) \(\chi_{1053}(88,\cdot)\) \(\chi_{1053}(121,\cdot)\) \(\chi_{1053}(205,\cdot)\) \(\chi_{1053}(238,\cdot)\) \(\chi_{1053}(322,\cdot)\) \(\chi_{1053}(355,\cdot)\) \(\chi_{1053}(439,\cdot)\) \(\chi_{1053}(472,\cdot)\) \(\chi_{1053}(556,\cdot)\) \(\chi_{1053}(589,\cdot)\) \(\chi_{1053}(673,\cdot)\) \(\chi_{1053}(706,\cdot)\) \(\chi_{1053}(790,\cdot)\) \(\chi_{1053}(823,\cdot)\) \(\chi_{1053}(907,\cdot)\) \(\chi_{1053}(940,\cdot)\) \(\chi_{1053}(1024,\cdot)\)
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
\((326,730)\) → \((e\left(\frac{1}{27}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 1053 }(4, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) |
sage: chi.jacobi_sum(n)