from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(691, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([6]))
pari: [g,chi] = znchar(Mod(329,691))
Basic properties
| Modulus: | \(691\) | |
| Conductor: | \(691\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(23\) | 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 691.h
\(\chi_{691}(20,\cdot)\) \(\chi_{691}(51,\cdot)\) \(\chi_{691}(195,\cdot)\) \(\chi_{691}(271,\cdot)\) \(\chi_{691}(310,\cdot)\) \(\chi_{691}(311,\cdot)\) \(\chi_{691}(329,\cdot)\) \(\chi_{691}(333,\cdot)\) \(\chi_{691}(361,\cdot)\) \(\chi_{691}(379,\cdot)\) \(\chi_{691}(399,\cdot)\) \(\chi_{691}(400,\cdot)\) \(\chi_{691}(413,\cdot)\) \(\chi_{691}(441,\cdot)\) \(\chi_{691}(445,\cdot)\) \(\chi_{691}(528,\cdot)\) \(\chi_{691}(583,\cdot)\) \(\chi_{691}(604,\cdot)\) \(\chi_{691}(608,\cdot)\) \(\chi_{691}(659,\cdot)\) \(\chi_{691}(670,\cdot)\) \(\chi_{691}(672,\cdot)\)
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: | \(\Q(\zeta_{23})\) |
| Fixed field: | Number field defined by a degree 23 polynomial |
Values on generators
\(3\) → \(e\left(\frac{3}{23}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 691 }(329, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) |
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)