from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1053, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([14,45]))
pari: [g,chi] = znchar(Mod(751,1053))
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.bz
\(\chi_{1053}(43,\cdot)\) \(\chi_{1053}(49,\cdot)\) \(\chi_{1053}(160,\cdot)\) \(\chi_{1053}(166,\cdot)\) \(\chi_{1053}(277,\cdot)\) \(\chi_{1053}(283,\cdot)\) \(\chi_{1053}(394,\cdot)\) \(\chi_{1053}(400,\cdot)\) \(\chi_{1053}(511,\cdot)\) \(\chi_{1053}(517,\cdot)\) \(\chi_{1053}(628,\cdot)\) \(\chi_{1053}(634,\cdot)\) \(\chi_{1053}(745,\cdot)\) \(\chi_{1053}(751,\cdot)\) \(\chi_{1053}(862,\cdot)\) \(\chi_{1053}(868,\cdot)\) \(\chi_{1053}(979,\cdot)\) \(\chi_{1053}(985,\cdot)\)
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
\((326,730)\) → \((e\left(\frac{7}{27}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 1053 }(751, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) |
sage: chi.jacobi_sum(n)