from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(350, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([51,50]))
pari: [g,chi] = znchar(Mod(47,350))
Basic properties
| Modulus: | \(350\) | |
| Conductor: | \(175\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{175}(47,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 350.x
\(\chi_{350}(3,\cdot)\) \(\chi_{350}(17,\cdot)\) \(\chi_{350}(33,\cdot)\) \(\chi_{350}(47,\cdot)\) \(\chi_{350}(73,\cdot)\) \(\chi_{350}(87,\cdot)\) \(\chi_{350}(103,\cdot)\) \(\chi_{350}(117,\cdot)\) \(\chi_{350}(173,\cdot)\) \(\chi_{350}(187,\cdot)\) \(\chi_{350}(213,\cdot)\) \(\chi_{350}(227,\cdot)\) \(\chi_{350}(283,\cdot)\) \(\chi_{350}(297,\cdot)\) \(\chi_{350}(313,\cdot)\) \(\chi_{350}(327,\cdot)\)
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
\((127,101)\) → \((e\left(\frac{17}{20}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 350 }(47, a) \) | \(1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{30}\right)\) |
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)