from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(800, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,5,18]))
pari: [g,chi] = znchar(Mod(187,800))
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.cd
\(\chi_{800}(3,\cdot)\) \(\chi_{800}(27,\cdot)\) \(\chi_{800}(83,\cdot)\) \(\chi_{800}(163,\cdot)\) \(\chi_{800}(187,\cdot)\) \(\chi_{800}(267,\cdot)\) \(\chi_{800}(323,\cdot)\) \(\chi_{800}(347,\cdot)\) \(\chi_{800}(403,\cdot)\) \(\chi_{800}(427,\cdot)\) \(\chi_{800}(483,\cdot)\) \(\chi_{800}(563,\cdot)\) \(\chi_{800}(587,\cdot)\) \(\chi_{800}(667,\cdot)\) \(\chi_{800}(723,\cdot)\) \(\chi_{800}(747,\cdot)\)
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: | 40.40.386856262276681335905976320000000000000000000000000000000000000000000000000000000000000000000000.1 |
Values on generators
\((351,101,577)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{9}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 800 }(187, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{40}\right)\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{40}\right)\) |
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)