from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(403, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([25,32]))
pari: [g,chi] = znchar(Mod(214,403))
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.ca
\(\chi_{403}(7,\cdot)\) \(\chi_{403}(19,\cdot)\) \(\chi_{403}(41,\cdot)\) \(\chi_{403}(45,\cdot)\) \(\chi_{403}(59,\cdot)\) \(\chi_{403}(80,\cdot)\) \(\chi_{403}(102,\cdot)\) \(\chi_{403}(162,\cdot)\) \(\chi_{403}(206,\cdot)\) \(\chi_{403}(214,\cdot)\) \(\chi_{403}(258,\cdot)\) \(\chi_{403}(267,\cdot)\) \(\chi_{403}(288,\cdot)\) \(\chi_{403}(293,\cdot)\) \(\chi_{403}(297,\cdot)\) \(\chi_{403}(392,\cdot)\)
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
\((249,313)\) → \((e\left(\frac{5}{12}\right),e\left(\frac{8}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 403 }(214, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) |
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)