from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(800, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,25,4]))
pari: [g,chi] = znchar(Mod(779,800))
Basic properties
| Modulus: | \(800\) | |
| Conductor: | \(800\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 800.bz
\(\chi_{800}(19,\cdot)\) \(\chi_{800}(59,\cdot)\) \(\chi_{800}(139,\cdot)\) \(\chi_{800}(179,\cdot)\) \(\chi_{800}(219,\cdot)\) \(\chi_{800}(259,\cdot)\) \(\chi_{800}(339,\cdot)\) \(\chi_{800}(379,\cdot)\) \(\chi_{800}(419,\cdot)\) \(\chi_{800}(459,\cdot)\) \(\chi_{800}(539,\cdot)\) \(\chi_{800}(579,\cdot)\) \(\chi_{800}(619,\cdot)\) \(\chi_{800}(659,\cdot)\) \(\chi_{800}(739,\cdot)\) \(\chi_{800}(779,\cdot)\)
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: | 40.0.15474250491067253436239052800000000000000000000000000000000000000000000000000000000000000000000.1 |
Values on generators
\((351,101,577)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{1}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 800 }(779, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{40}\right)\) | \(i\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) |
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)