from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(548, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,12]))
pari: [g,chi] = znchar(Mod(171,548))
Basic properties
Modulus: | \(548\) | |
Conductor: | \(548\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(34\) | 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 548.k
\(\chi_{548}(59,\cdot)\) \(\chi_{548}(115,\cdot)\) \(\chi_{548}(119,\cdot)\) \(\chi_{548}(123,\cdot)\) \(\chi_{548}(171,\cdot)\) \(\chi_{548}(175,\cdot)\) \(\chi_{548}(187,\cdot)\) \(\chi_{548}(211,\cdot)\) \(\chi_{548}(259,\cdot)\) \(\chi_{548}(347,\cdot)\) \(\chi_{548}(407,\cdot)\) \(\chi_{548}(427,\cdot)\) \(\chi_{548}(467,\cdot)\) \(\chi_{548}(471,\cdot)\) \(\chi_{548}(483,\cdot)\) \(\chi_{548}(499,\cdot)\)
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: | \(\Q(\zeta_{17})\) |
Fixed field: | 34.0.4074534520224700001793556316626715832180494310341751036114303141235744313442304.1 |
Values on generators
\((275,277)\) → \((-1,e\left(\frac{6}{17}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 548 }(171, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{3}{17}\right)\) |
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)