from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(469, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,20]))
pari: [g,chi] = znchar(Mod(26,469))
Basic properties
| Modulus: | \(469\) | |
| Conductor: | \(469\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | 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 469.bj
\(\chi_{469}(10,\cdot)\) \(\chi_{469}(17,\cdot)\) \(\chi_{469}(26,\cdot)\) \(\chi_{469}(33,\cdot)\) \(\chi_{469}(47,\cdot)\) \(\chi_{469}(73,\cdot)\) \(\chi_{469}(103,\cdot)\) \(\chi_{469}(138,\cdot)\) \(\chi_{469}(150,\cdot)\) \(\chi_{469}(157,\cdot)\) \(\chi_{469}(194,\cdot)\) \(\chi_{469}(199,\cdot)\) \(\chi_{469}(220,\cdot)\) \(\chi_{469}(222,\cdot)\) \(\chi_{469}(236,\cdot)\) \(\chi_{469}(255,\cdot)\) \(\chi_{469}(257,\cdot)\) \(\chi_{469}(374,\cdot)\) \(\chi_{469}(390,\cdot)\) \(\chi_{469}(451,\cdot)\)
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 66 polynomial |
Values on generators
\((269,337)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{10}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 469 }(26, a) \) | \(-1\) | \(1\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) |
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)