from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(950, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([57,10]))
pari: [g,chi] = znchar(Mod(863,950))
Basic properties
| Modulus: | \(950\) | |
| Conductor: | \(475\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{475}(388,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 950.bd
\(\chi_{950}(27,\cdot)\) \(\chi_{950}(103,\cdot)\) \(\chi_{950}(183,\cdot)\) \(\chi_{950}(217,\cdot)\) \(\chi_{950}(297,\cdot)\) \(\chi_{950}(373,\cdot)\) \(\chi_{950}(483,\cdot)\) \(\chi_{950}(487,\cdot)\) \(\chi_{950}(563,\cdot)\) \(\chi_{950}(597,\cdot)\) \(\chi_{950}(673,\cdot)\) \(\chi_{950}(677,\cdot)\) \(\chi_{950}(753,\cdot)\) \(\chi_{950}(787,\cdot)\) \(\chi_{950}(863,\cdot)\) \(\chi_{950}(867,\cdot)\)
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
\((77,401)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 950 }(863, a) \) | \(1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(-i\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{15}\right)\) |
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)