from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(496, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,15,32]))
pari: [g,chi] = znchar(Mod(245,496))
Basic properties
| Modulus: | \(496\) | |
| Conductor: | \(496\) | 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 496.bs
\(\chi_{496}(45,\cdot)\) \(\chi_{496}(69,\cdot)\) \(\chi_{496}(133,\cdot)\) \(\chi_{496}(165,\cdot)\) \(\chi_{496}(173,\cdot)\) \(\chi_{496}(205,\cdot)\) \(\chi_{496}(237,\cdot)\) \(\chi_{496}(245,\cdot)\) \(\chi_{496}(293,\cdot)\) \(\chi_{496}(317,\cdot)\) \(\chi_{496}(381,\cdot)\) \(\chi_{496}(413,\cdot)\) \(\chi_{496}(421,\cdot)\) \(\chi_{496}(453,\cdot)\) \(\chi_{496}(485,\cdot)\) \(\chi_{496}(493,\cdot)\)
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
\((63,373,65)\) → \((1,i,e\left(\frac{8}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 496 }(245, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) |
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)