from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(851, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([4,0]))
pari: [g,chi] = znchar(Mod(556,851))
Basic properties
Modulus: | \(851\) | |
Conductor: | \(23\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(11\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{23}(4,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 851.l
\(\chi_{851}(75,\cdot)\) \(\chi_{851}(186,\cdot)\) \(\chi_{851}(223,\cdot)\) \(\chi_{851}(334,\cdot)\) \(\chi_{851}(371,\cdot)\) \(\chi_{851}(445,\cdot)\) \(\chi_{851}(519,\cdot)\) \(\chi_{851}(556,\cdot)\) \(\chi_{851}(593,\cdot)\) \(\chi_{851}(630,\cdot)\)
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: | \(\Q(\zeta_{23})^+\) |
Values on generators
\((741,668)\) → \((e\left(\frac{2}{11}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 851 }(556, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) |
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)