from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(527, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([15,34]))
pari: [g,chi] = znchar(Mod(115,527))
Basic properties
| Modulus: | \(527\) | |
| Conductor: | \(527\) | 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 527.bg
\(\chi_{527}(13,\cdot)\) \(\chi_{527}(21,\cdot)\) \(\chi_{527}(55,\cdot)\) \(\chi_{527}(106,\cdot)\) \(\chi_{527}(115,\cdot)\) \(\chi_{527}(166,\cdot)\) \(\chi_{527}(208,\cdot)\) \(\chi_{527}(234,\cdot)\) \(\chi_{527}(251,\cdot)\) \(\chi_{527}(259,\cdot)\) \(\chi_{527}(327,\cdot)\) \(\chi_{527}(344,\cdot)\) \(\chi_{527}(353,\cdot)\) \(\chi_{527}(446,\cdot)\) \(\chi_{527}(455,\cdot)\) \(\chi_{527}(489,\cdot)\)
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
\((156,375)\) → \((i,e\left(\frac{17}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 527 }(115, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) |
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)