from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(805, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,22,42]))
pari: [g,chi] = znchar(Mod(289,805))
Basic properties
Modulus: | \(805\) | |
Conductor: | \(805\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 805.bm
\(\chi_{805}(4,\cdot)\) \(\chi_{805}(9,\cdot)\) \(\chi_{805}(39,\cdot)\) \(\chi_{805}(144,\cdot)\) \(\chi_{805}(179,\cdot)\) \(\chi_{805}(219,\cdot)\) \(\chi_{805}(284,\cdot)\) \(\chi_{805}(289,\cdot)\) \(\chi_{805}(324,\cdot)\) \(\chi_{805}(354,\cdot)\) \(\chi_{805}(394,\cdot)\) \(\chi_{805}(464,\cdot)\) \(\chi_{805}(499,\cdot)\) \(\chi_{805}(564,\cdot)\) \(\chi_{805}(604,\cdot)\) \(\chi_{805}(634,\cdot)\) \(\chi_{805}(639,\cdot)\) \(\chi_{805}(669,\cdot)\) \(\chi_{805}(739,\cdot)\) \(\chi_{805}(744,\cdot)\)
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
\((162,346,281)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{7}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
\( \chi_{ 805 }(289, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{25}{33}\right)\) |
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)