from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(69, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,13]))
pari: [g,chi] = znchar(Mod(67,69))
Basic properties
| Modulus: | \(69\) | |
| Conductor: | \(23\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{23}(21,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 69.f
\(\chi_{69}(7,\cdot)\) \(\chi_{69}(10,\cdot)\) \(\chi_{69}(19,\cdot)\) \(\chi_{69}(28,\cdot)\) \(\chi_{69}(34,\cdot)\) \(\chi_{69}(37,\cdot)\) \(\chi_{69}(40,\cdot)\) \(\chi_{69}(43,\cdot)\) \(\chi_{69}(61,\cdot)\) \(\chi_{69}(67,\cdot)\)
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
\((47,28)\) → \((1,e\left(\frac{13}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 69 }(67, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) |
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)