from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(388, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,35]))
pari: [g,chi] = znchar(Mod(3,388))
Basic properties
Modulus: | \(388\) | |
Conductor: | \(388\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | 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 388.v
\(\chi_{388}(3,\cdot)\) \(\chi_{388}(11,\cdot)\) \(\chi_{388}(31,\cdot)\) \(\chi_{388}(95,\cdot)\) \(\chi_{388}(99,\cdot)\) \(\chi_{388}(163,\cdot)\) \(\chi_{388}(183,\cdot)\) \(\chi_{388}(191,\cdot)\) \(\chi_{388}(219,\cdot)\) \(\chi_{388}(243,\cdot)\) \(\chi_{388}(247,\cdot)\) \(\chi_{388}(259,\cdot)\) \(\chi_{388}(323,\cdot)\) \(\chi_{388}(335,\cdot)\) \(\chi_{388}(339,\cdot)\) \(\chi_{388}(363,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((195,5)\) → \((-1,e\left(\frac{35}{48}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 388 }(3, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{31}{48}\right)\) |
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)