from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(704, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,5,8]))
pari: [g,chi] = znchar(Mod(103,704))
Basic properties
| Modulus: | \(704\) | |
| Conductor: | \(352\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{352}(59,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 704.bh
\(\chi_{704}(71,\cdot)\) \(\chi_{704}(103,\cdot)\) \(\chi_{704}(119,\cdot)\) \(\chi_{704}(135,\cdot)\) \(\chi_{704}(247,\cdot)\) \(\chi_{704}(279,\cdot)\) \(\chi_{704}(295,\cdot)\) \(\chi_{704}(311,\cdot)\) \(\chi_{704}(423,\cdot)\) \(\chi_{704}(455,\cdot)\) \(\chi_{704}(471,\cdot)\) \(\chi_{704}(487,\cdot)\) \(\chi_{704}(599,\cdot)\) \(\chi_{704}(631,\cdot)\) \(\chi_{704}(647,\cdot)\) \(\chi_{704}(663,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.0.96430685261162182749113906515642066253992366248338958954046471967872161601814528.1 |
Values on generators
\((639,133,321)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{1}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 704 }(103, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(i\) |
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)