from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(403, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,4]))
pari: [g,chi] = znchar(Mod(226,403))
Basic properties
| Modulus: | \(403\) | |
| Conductor: | \(403\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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 403.cg
\(\chi_{403}(18,\cdot)\) \(\chi_{403}(112,\cdot)\) \(\chi_{403}(138,\cdot)\) \(\chi_{403}(164,\cdot)\) \(\chi_{403}(174,\cdot)\) \(\chi_{403}(200,\cdot)\) \(\chi_{403}(226,\cdot)\) \(\chi_{403}(255,\cdot)\) \(\chi_{403}(268,\cdot)\) \(\chi_{403}(307,\cdot)\) \(\chi_{403}(317,\cdot)\) \(\chi_{403}(320,\cdot)\) \(\chi_{403}(330,\cdot)\) \(\chi_{403}(359,\cdot)\) \(\chi_{403}(369,\cdot)\) \(\chi_{403}(382,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((249,313)\) → \((-i,e\left(\frac{1}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 403 }(226, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{47}{60}\right)\) |
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)