from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(851, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([18,11]))
pari: [g,chi] = znchar(Mod(586,851))
Basic properties
| Modulus: | \(851\) | |
| Conductor: | \(851\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | 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 851.y
\(\chi_{851}(43,\cdot)\) \(\chi_{851}(80,\cdot)\) \(\chi_{851}(191,\cdot)\) \(\chi_{851}(228,\cdot)\) \(\chi_{851}(290,\cdot)\) \(\chi_{851}(327,\cdot)\) \(\chi_{851}(339,\cdot)\) \(\chi_{851}(364,\cdot)\) \(\chi_{851}(401,\cdot)\) \(\chi_{851}(475,\cdot)\) \(\chi_{851}(549,\cdot)\) \(\chi_{851}(586,\cdot)\) \(\chi_{851}(635,\cdot)\) \(\chi_{851}(672,\cdot)\) \(\chi_{851}(697,\cdot)\) \(\chi_{851}(709,\cdot)\) \(\chi_{851}(734,\cdot)\) \(\chi_{851}(746,\cdot)\) \(\chi_{851}(820,\cdot)\) \(\chi_{851}(845,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | 44.44.8774573313493339673130485496498509268977103264640582594368355759183398191746496649773237189225908287319938613.1 |
Values on generators
\((741,668)\) → \((e\left(\frac{9}{22}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 851 }(586, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) |
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)