from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(268, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,3]))
pari: [g,chi] = znchar(Mod(43,268))
Basic properties
| Modulus: | \(268\) | |
| Conductor: | \(268\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | 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 268.j
\(\chi_{268}(3,\cdot)\) \(\chi_{268}(27,\cdot)\) \(\chi_{268}(43,\cdot)\) \(\chi_{268}(75,\cdot)\) \(\chi_{268}(119,\cdot)\) \(\chi_{268}(139,\cdot)\) \(\chi_{268}(179,\cdot)\) \(\chi_{268}(187,\cdot)\) \(\chi_{268}(243,\cdot)\) \(\chi_{268}(259,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
| Fixed field: | 22.22.933750314983937236210361829997141120063111168.1 |
Values on generators
\((135,69)\) → \((-1,e\left(\frac{3}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 268 }(43, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) |
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)