from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(729, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([4]))
pari: [g,chi] = znchar(Mod(136,729))
Basic properties
| Modulus: | \(729\) | |
| Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(27\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{81}(16,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 729.g
\(\chi_{729}(28,\cdot)\) \(\chi_{729}(55,\cdot)\) \(\chi_{729}(109,\cdot)\) \(\chi_{729}(136,\cdot)\) \(\chi_{729}(190,\cdot)\) \(\chi_{729}(217,\cdot)\) \(\chi_{729}(271,\cdot)\) \(\chi_{729}(298,\cdot)\) \(\chi_{729}(352,\cdot)\) \(\chi_{729}(379,\cdot)\) \(\chi_{729}(433,\cdot)\) \(\chi_{729}(460,\cdot)\) \(\chi_{729}(514,\cdot)\) \(\chi_{729}(541,\cdot)\) \(\chi_{729}(595,\cdot)\) \(\chi_{729}(622,\cdot)\) \(\chi_{729}(676,\cdot)\) \(\chi_{729}(703,\cdot)\)
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
\(2\) → \(e\left(\frac{2}{27}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 729 }(136, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)