from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(812, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,14,27]))
pari: [g,chi] = znchar(Mod(303,812))
Basic properties
| Modulus: | \(812\) | |
| Conductor: | \(812\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | 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 812.br
\(\chi_{812}(51,\cdot)\) \(\chi_{812}(67,\cdot)\) \(\chi_{812}(151,\cdot)\) \(\chi_{812}(179,\cdot)\) \(\chi_{812}(207,\cdot)\) \(\chi_{812}(303,\cdot)\) \(\chi_{812}(415,\cdot)\) \(\chi_{812}(499,\cdot)\) \(\chi_{812}(515,\cdot)\) \(\chi_{812}(527,\cdot)\) \(\chi_{812}(555,\cdot)\) \(\chi_{812}(767,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((407,465,785)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{9}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 812 }(303, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) |
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)