from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(175, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([51,50]))
pari: [g,chi] = znchar(Mod(47,175))
Basic properties
| Modulus: | \(175\) | |
| Conductor: | \(175\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 175.x
\(\chi_{175}(3,\cdot)\) \(\chi_{175}(12,\cdot)\) \(\chi_{175}(17,\cdot)\) \(\chi_{175}(33,\cdot)\) \(\chi_{175}(38,\cdot)\) \(\chi_{175}(47,\cdot)\) \(\chi_{175}(52,\cdot)\) \(\chi_{175}(73,\cdot)\) \(\chi_{175}(87,\cdot)\) \(\chi_{175}(103,\cdot)\) \(\chi_{175}(108,\cdot)\) \(\chi_{175}(117,\cdot)\) \(\chi_{175}(122,\cdot)\) \(\chi_{175}(138,\cdot)\) \(\chi_{175}(152,\cdot)\) \(\chi_{175}(173,\cdot)\)
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
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 175 }(47, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) |
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)