from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(469, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,58]))
pari: [g,chi] = znchar(Mod(307,469))
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.bl
\(\chi_{469}(6,\cdot)\) \(\chi_{469}(55,\cdot)\) \(\chi_{469}(83,\cdot)\) \(\chi_{469}(90,\cdot)\) \(\chi_{469}(132,\cdot)\) \(\chi_{469}(153,\cdot)\) \(\chi_{469}(160,\cdot)\) \(\chi_{469}(167,\cdot)\) \(\chi_{469}(181,\cdot)\) \(\chi_{469}(188,\cdot)\) \(\chi_{469}(237,\cdot)\) \(\chi_{469}(272,\cdot)\) \(\chi_{469}(307,\cdot)\) \(\chi_{469}(328,\cdot)\) \(\chi_{469}(356,\cdot)\) \(\chi_{469}(370,\cdot)\) \(\chi_{469}(384,\cdot)\) \(\chi_{469}(391,\cdot)\) \(\chi_{469}(412,\cdot)\) \(\chi_{469}(419,\cdot)\)
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
\((269,337)\) → \((-1,e\left(\frac{29}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 469 }(307, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) |
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)