from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(352, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,15,36]))
pari: [g,chi] = znchar(Mod(259,352))
Basic properties
| Modulus: | \(352\) | |
| Conductor: | \(352\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | 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 352.bc
\(\chi_{352}(19,\cdot)\) \(\chi_{352}(35,\cdot)\) \(\chi_{352}(51,\cdot)\) \(\chi_{352}(83,\cdot)\) \(\chi_{352}(107,\cdot)\) \(\chi_{352}(123,\cdot)\) \(\chi_{352}(139,\cdot)\) \(\chi_{352}(171,\cdot)\) \(\chi_{352}(195,\cdot)\) \(\chi_{352}(211,\cdot)\) \(\chi_{352}(227,\cdot)\) \(\chi_{352}(259,\cdot)\) \(\chi_{352}(283,\cdot)\) \(\chi_{352}(299,\cdot)\) \(\chi_{352}(315,\cdot)\) \(\chi_{352}(347,\cdot)\)
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: | 40.40.1411841662908675517629776705295515492024702234241930698046194396081616318012166504448.1 |
Values on generators
\((287,133,321)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{9}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 352 }(259, a) \) | \(1\) | \(1\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) |
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)