from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(671, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([42,49]))
pari: [g,chi] = znchar(Mod(7,671))
Basic properties
| Modulus: | \(671\) | |
| Conductor: | \(671\) | 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 671.cu
\(\chi_{671}(7,\cdot)\) \(\chi_{671}(63,\cdot)\) \(\chi_{671}(96,\cdot)\) \(\chi_{671}(189,\cdot)\) \(\chi_{671}(193,\cdot)\) \(\chi_{671}(200,\cdot)\) \(\chi_{671}(226,\cdot)\) \(\chi_{671}(288,\cdot)\) \(\chi_{671}(299,\cdot)\) \(\chi_{671}(359,\cdot)\) \(\chi_{671}(458,\cdot)\) \(\chi_{671}(514,\cdot)\) \(\chi_{671}(547,\cdot)\) \(\chi_{671}(567,\cdot)\) \(\chi_{671}(579,\cdot)\) \(\chi_{671}(600,\cdot)\)
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
\((123,551)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{49}{60}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 671 }(7, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(-1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) |
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)