from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(23, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([13]))
pari: [g,chi] = znchar(Mod(21,23))
Basic properties
| Modulus: | \(23\) | |
| Conductor: | \(23\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | 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 23.d
\(\chi_{23}(5,\cdot)\) \(\chi_{23}(7,\cdot)\) \(\chi_{23}(10,\cdot)\) \(\chi_{23}(11,\cdot)\) \(\chi_{23}(14,\cdot)\) \(\chi_{23}(15,\cdot)\) \(\chi_{23}(17,\cdot)\) \(\chi_{23}(19,\cdot)\) \(\chi_{23}(20,\cdot)\) \(\chi_{23}(21,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
| Fixed field: | Number field defined by a degree 22 polynomial |
Values on generators
\(5\) → \(e\left(\frac{13}{22}\right)\)
Values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 23 }(21, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) |
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)