from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(79, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([1]))
pari: [g,chi] = znchar(Mod(3,79))
Basic properties
| Modulus: | \(79\) | |
| Conductor: | \(79\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(78\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 79.h
\(\chi_{79}(3,\cdot)\) \(\chi_{79}(6,\cdot)\) \(\chi_{79}(7,\cdot)\) \(\chi_{79}(28,\cdot)\) \(\chi_{79}(29,\cdot)\) \(\chi_{79}(30,\cdot)\) \(\chi_{79}(34,\cdot)\) \(\chi_{79}(35,\cdot)\) \(\chi_{79}(37,\cdot)\) \(\chi_{79}(39,\cdot)\) \(\chi_{79}(43,\cdot)\) \(\chi_{79}(47,\cdot)\) \(\chi_{79}(48,\cdot)\) \(\chi_{79}(53,\cdot)\) \(\chi_{79}(54,\cdot)\) \(\chi_{79}(59,\cdot)\) \(\chi_{79}(60,\cdot)\) \(\chi_{79}(63,\cdot)\) \(\chi_{79}(66,\cdot)\) \(\chi_{79}(68,\cdot)\) \(\chi_{79}(70,\cdot)\) \(\chi_{79}(74,\cdot)\) \(\chi_{79}(75,\cdot)\) \(\chi_{79}(77,\cdot)\)
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 78 polynomial |
Values on generators
\(3\) → \(e\left(\frac{1}{78}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 79 }(3, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{34}{39}\right)\) |
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)