from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(464, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,21,4]))
pari: [g,chi] = znchar(Mod(45,464))
Basic properties
Modulus: | \(464\) | |
Conductor: | \(464\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | 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 464.bi
\(\chi_{464}(45,\cdot)\) \(\chi_{464}(53,\cdot)\) \(\chi_{464}(141,\cdot)\) \(\chi_{464}(165,\cdot)\) \(\chi_{464}(181,\cdot)\) \(\chi_{464}(197,\cdot)\) \(\chi_{464}(277,\cdot)\) \(\chi_{464}(285,\cdot)\) \(\chi_{464}(373,\cdot)\) \(\chi_{464}(397,\cdot)\) \(\chi_{464}(413,\cdot)\) \(\chi_{464}(429,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((175,117,321)\) → \((1,-i,e\left(\frac{1}{7}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 464 }(45, a) \) | \(1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{5}{28}\right)\) |
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)