from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(47, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([40]))
pari: [g,chi] = znchar(Mod(9,47))
Basic properties
| Modulus: | \(47\) | |
| Conductor: | \(47\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(23\) | 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 47.c
\(\chi_{47}(2,\cdot)\) \(\chi_{47}(3,\cdot)\) \(\chi_{47}(4,\cdot)\) \(\chi_{47}(6,\cdot)\) \(\chi_{47}(7,\cdot)\) \(\chi_{47}(8,\cdot)\) \(\chi_{47}(9,\cdot)\) \(\chi_{47}(12,\cdot)\) \(\chi_{47}(14,\cdot)\) \(\chi_{47}(16,\cdot)\) \(\chi_{47}(17,\cdot)\) \(\chi_{47}(18,\cdot)\) \(\chi_{47}(21,\cdot)\) \(\chi_{47}(24,\cdot)\) \(\chi_{47}(25,\cdot)\) \(\chi_{47}(27,\cdot)\) \(\chi_{47}(28,\cdot)\) \(\chi_{47}(32,\cdot)\) \(\chi_{47}(34,\cdot)\) \(\chi_{47}(36,\cdot)\) \(\chi_{47}(37,\cdot)\) \(\chi_{47}(42,\cdot)\)
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: | \(\Q(\zeta_{23})\) |
| Fixed field: | \(\Q(\zeta_{47})^+\) |
Values on generators
\(5\) → \(e\left(\frac{20}{23}\right)\)
Values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 47 }(9, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) |
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)