from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8033, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([3,7]))
pari: [g,chi] = znchar(Mod(671,8033))
Basic properties
Modulus: | \(8033\) | |
Conductor: | \(8033\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8033.bg
\(\chi_{8033}(671,\cdot)\) \(\chi_{8033}(992,\cdot)\) \(\chi_{8033}(1269,\cdot)\) \(\chi_{8033}(1546,\cdot)\) \(\chi_{8033}(2333,\cdot)\) \(\chi_{8033}(3485,\cdot)\) \(\chi_{8033}(3718,\cdot)\) \(\chi_{8033}(3995,\cdot)\) \(\chi_{8033}(4272,\cdot)\) \(\chi_{8033}(5978,\cdot)\) \(\chi_{8033}(6211,\cdot)\) \(\chi_{8033}(7640,\cdot)\)
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
\((5541,1944)\) → \((e\left(\frac{1}{14}\right),e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 8033 }(671, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) |
sage: chi.jacobi_sum(n)