from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(299, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,8]))
pari: [g,chi] = znchar(Mod(246,299))
Basic properties
Modulus: | \(299\) | |
Conductor: | \(299\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | 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 299.p
\(\chi_{299}(12,\cdot)\) \(\chi_{299}(25,\cdot)\) \(\chi_{299}(64,\cdot)\) \(\chi_{299}(77,\cdot)\) \(\chi_{299}(142,\cdot)\) \(\chi_{299}(220,\cdot)\) \(\chi_{299}(233,\cdot)\) \(\chi_{299}(246,\cdot)\) \(\chi_{299}(259,\cdot)\) \(\chi_{299}(285,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
Fixed field: | Number field defined by a degree 22 polynomial |
Values on generators
\((93,235)\) → \((-1,e\left(\frac{4}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 299 }(246, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) |
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)