from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(197, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([5]))
pari: [g,chi] = znchar(Mod(113,197))
Basic properties
| Modulus: | \(197\) | |
| Conductor: | \(197\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | 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 197.f
\(\chi_{197}(20,\cdot)\) \(\chi_{197}(68,\cdot)\) \(\chi_{197}(69,\cdot)\) \(\chi_{197}(77,\cdot)\) \(\chi_{197}(84,\cdot)\) \(\chi_{197}(87,\cdot)\) \(\chi_{197}(110,\cdot)\) \(\chi_{197}(113,\cdot)\) \(\chi_{197}(120,\cdot)\) \(\chi_{197}(128,\cdot)\) \(\chi_{197}(129,\cdot)\) \(\chi_{197}(177,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\(2\) → \(e\left(\frac{5}{28}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 197 }(113, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(-1\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) |
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)