from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(812, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,14,18]))
pari: [g,chi] = znchar(Mod(471,812))
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.bn
\(\chi_{812}(23,\cdot)\) \(\chi_{812}(107,\cdot)\) \(\chi_{812}(123,\cdot)\) \(\chi_{812}(219,\cdot)\) \(\chi_{812}(431,\cdot)\) \(\chi_{812}(459,\cdot)\) \(\chi_{812}(471,\cdot)\) \(\chi_{812}(487,\cdot)\) \(\chi_{812}(571,\cdot)\) \(\chi_{812}(683,\cdot)\) \(\chi_{812}(779,\cdot)\) \(\chi_{812}(807,\cdot)\)
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
\((407,465,785)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{3}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 812 }(471, a) \) | \(-1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) |
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)