from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(800, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,15,16]))
pari: [g,chi] = znchar(Mod(381,800))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 800.ca
\(\chi_{800}(21,\cdot)\) \(\chi_{800}(61,\cdot)\) \(\chi_{800}(141,\cdot)\) \(\chi_{800}(181,\cdot)\) \(\chi_{800}(221,\cdot)\) \(\chi_{800}(261,\cdot)\) \(\chi_{800}(341,\cdot)\) \(\chi_{800}(381,\cdot)\) \(\chi_{800}(421,\cdot)\) \(\chi_{800}(461,\cdot)\) \(\chi_{800}(541,\cdot)\) \(\chi_{800}(581,\cdot)\) \(\chi_{800}(621,\cdot)\) \(\chi_{800}(661,\cdot)\) \(\chi_{800}(741,\cdot)\) \(\chi_{800}(781,\cdot)\)
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: | Number field defined by a degree 40 polynomial |
Values on generators
\((351,101,577)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{2}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 800 }(381, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{40}\right)\) | \(-i\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) |
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)