from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(67, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([8]))
pari: [g,chi] = znchar(Mod(55,67))
Basic properties
Modulus: | \(67\) | |
Conductor: | \(67\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | 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 67.g
\(\chi_{67}(4,\cdot)\) \(\chi_{67}(6,\cdot)\) \(\chi_{67}(10,\cdot)\) \(\chi_{67}(16,\cdot)\) \(\chi_{67}(17,\cdot)\) \(\chi_{67}(19,\cdot)\) \(\chi_{67}(21,\cdot)\) \(\chi_{67}(23,\cdot)\) \(\chi_{67}(26,\cdot)\) \(\chi_{67}(33,\cdot)\) \(\chi_{67}(35,\cdot)\) \(\chi_{67}(36,\cdot)\) \(\chi_{67}(39,\cdot)\) \(\chi_{67}(47,\cdot)\) \(\chi_{67}(49,\cdot)\) \(\chi_{67}(54,\cdot)\) \(\chi_{67}(55,\cdot)\) \(\chi_{67}(56,\cdot)\) \(\chi_{67}(60,\cdot)\) \(\chi_{67}(65,\cdot)\)
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 33 polynomial |
Values on generators
\(2\) → \(e\left(\frac{4}{33}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 67 }(55, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) |
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)