from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(469, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([22,30]))
pari: [g,chi] = znchar(Mod(226,469))
Basic properties
Modulus: | \(469\) | |
Conductor: | \(469\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | 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 469.ba
\(\chi_{469}(9,\cdot)\) \(\chi_{469}(25,\cdot)\) \(\chi_{469}(81,\cdot)\) \(\chi_{469}(107,\cdot)\) \(\chi_{469}(149,\cdot)\) \(\chi_{469}(156,\cdot)\) \(\chi_{469}(158,\cdot)\) \(\chi_{469}(193,\cdot)\) \(\chi_{469}(198,\cdot)\) \(\chi_{469}(226,\cdot)\) \(\chi_{469}(263,\cdot)\) \(\chi_{469}(277,\cdot)\) \(\chi_{469}(282,\cdot)\) \(\chi_{469}(359,\cdot)\) \(\chi_{469}(375,\cdot)\) \(\chi_{469}(394,\cdot)\) \(\chi_{469}(417,\cdot)\) \(\chi_{469}(424,\cdot)\) \(\chi_{469}(464,\cdot)\) \(\chi_{469}(466,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
Fixed field: | Number field defined by a degree 33 polynomial |
Values on generators
\((269,337)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{5}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
\( \chi_{ 469 }(226, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) |
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)