from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(79, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([4]))
pari: [g,chi] = znchar(Mod(2,79))
Basic properties
Modulus: | \(79\) | |
Conductor: | \(79\) | 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 79.g
\(\chi_{79}(2,\cdot)\) \(\chi_{79}(4,\cdot)\) \(\chi_{79}(5,\cdot)\) \(\chi_{79}(9,\cdot)\) \(\chi_{79}(11,\cdot)\) \(\chi_{79}(13,\cdot)\) \(\chi_{79}(16,\cdot)\) \(\chi_{79}(19,\cdot)\) \(\chi_{79}(20,\cdot)\) \(\chi_{79}(25,\cdot)\) \(\chi_{79}(26,\cdot)\) \(\chi_{79}(31,\cdot)\) \(\chi_{79}(32,\cdot)\) \(\chi_{79}(36,\cdot)\) \(\chi_{79}(40,\cdot)\) \(\chi_{79}(42,\cdot)\) \(\chi_{79}(44,\cdot)\) \(\chi_{79}(45,\cdot)\) \(\chi_{79}(49,\cdot)\) \(\chi_{79}(50,\cdot)\) \(\chi_{79}(51,\cdot)\) \(\chi_{79}(72,\cdot)\) \(\chi_{79}(73,\cdot)\) \(\chi_{79}(76,\cdot)\)
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
\(3\) → \(e\left(\frac{2}{39}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 79 }(2, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{19}{39}\right)\) |
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)