from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(613, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([28]))
pari: [g,chi] = znchar(Mod(430,613))
Basic properties
| Modulus: | \(613\) | |
| Conductor: | \(613\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(17\) | 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 613.h
\(\chi_{613}(37,\cdot)\) \(\chi_{613}(116,\cdot)\) \(\chi_{613}(143,\cdot)\) \(\chi_{613}(171,\cdot)\) \(\chi_{613}(190,\cdot)\) \(\chi_{613}(197,\cdot)\) \(\chi_{613}(198,\cdot)\) \(\chi_{613}(220,\cdot)\) \(\chi_{613}(227,\cdot)\) \(\chi_{613}(287,\cdot)\) \(\chi_{613}(387,\cdot)\) \(\chi_{613}(430,\cdot)\) \(\chi_{613}(546,\cdot)\) \(\chi_{613}(583,\cdot)\) \(\chi_{613}(585,\cdot)\) \(\chi_{613}(586,\cdot)\)
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: | \(\Q(\zeta_{17})\) |
| Fixed field: | Number field defined by a degree 17 polynomial |
Values on generators
\(2\) → \(e\left(\frac{14}{17}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 613 }(430, a) \) | \(1\) | \(1\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) |
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)