from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(715, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,42,35]))
pari: [g,chi] = znchar(Mod(128,715))
Basic properties
| Modulus: | \(715\) | |
| Conductor: | \(715\) | 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 715.cr
\(\chi_{715}(2,\cdot)\) \(\chi_{715}(63,\cdot)\) \(\chi_{715}(128,\cdot)\) \(\chi_{715}(162,\cdot)\) \(\chi_{715}(193,\cdot)\) \(\chi_{715}(227,\cdot)\) \(\chi_{715}(228,\cdot)\) \(\chi_{715}(292,\cdot)\) \(\chi_{715}(293,\cdot)\) \(\chi_{715}(327,\cdot)\) \(\chi_{715}(358,\cdot)\) \(\chi_{715}(392,\cdot)\) \(\chi_{715}(453,\cdot)\) \(\chi_{715}(457,\cdot)\) \(\chi_{715}(552,\cdot)\) \(\chi_{715}(618,\cdot)\)
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
\((287,651,496)\) → \((-i,e\left(\frac{7}{10}\right),e\left(\frac{7}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(14\) | \(16\) |
| \( \chi_{ 715 }(128, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(i\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) |
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)