from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(169, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([40]))
pari: [g,chi] = znchar(Mod(165,169))
Basic properties
| Modulus: | \(169\) | |
| Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(39\) | 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 169.i
\(\chi_{169}(3,\cdot)\) \(\chi_{169}(9,\cdot)\) \(\chi_{169}(16,\cdot)\) \(\chi_{169}(29,\cdot)\) \(\chi_{169}(35,\cdot)\) \(\chi_{169}(42,\cdot)\) \(\chi_{169}(48,\cdot)\) \(\chi_{169}(55,\cdot)\) \(\chi_{169}(61,\cdot)\) \(\chi_{169}(68,\cdot)\) \(\chi_{169}(74,\cdot)\) \(\chi_{169}(81,\cdot)\) \(\chi_{169}(87,\cdot)\) \(\chi_{169}(94,\cdot)\) \(\chi_{169}(100,\cdot)\) \(\chi_{169}(107,\cdot)\) \(\chi_{169}(113,\cdot)\) \(\chi_{169}(120,\cdot)\) \(\chi_{169}(126,\cdot)\) \(\chi_{169}(133,\cdot)\) \(\chi_{169}(139,\cdot)\) \(\chi_{169}(152,\cdot)\) \(\chi_{169}(159,\cdot)\) \(\chi_{169}(165,\cdot)\)
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: | $\Q(\zeta_{39})$ |
| Fixed field: | Number field defined by a degree 39 polynomial |
Values on generators
\(2\) → \(e\left(\frac{20}{39}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 169 }(165, a) \) | \(1\) | \(1\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) |
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)