from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(704, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,15,12]))
pari: [g,chi] = znchar(Mod(393,704))
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 \(\chi_{352}(349,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 704.bi
\(\chi_{704}(41,\cdot)\) \(\chi_{704}(57,\cdot)\) \(\chi_{704}(73,\cdot)\) \(\chi_{704}(105,\cdot)\) \(\chi_{704}(217,\cdot)\) \(\chi_{704}(233,\cdot)\) \(\chi_{704}(249,\cdot)\) \(\chi_{704}(281,\cdot)\) \(\chi_{704}(393,\cdot)\) \(\chi_{704}(409,\cdot)\) \(\chi_{704}(425,\cdot)\) \(\chi_{704}(457,\cdot)\) \(\chi_{704}(569,\cdot)\) \(\chi_{704}(585,\cdot)\) \(\chi_{704}(601,\cdot)\) \(\chi_{704}(633,\cdot)\)
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: | 40.0.1411841662908675517629776705295515492024702234241930698046194396081616318012166504448.1 |
Values on generators
\((639,133,321)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{3}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 704 }(393, a) \) | \(-1\) | \(1\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(i\) |
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)