from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(485, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,17]))
pari: [g,chi] = znchar(Mod(69,485))
Basic properties
Modulus: | \(485\) | |
Conductor: | \(485\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(32\) | 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 485.bi
\(\chi_{485}(19,\cdot)\) \(\chi_{485}(34,\cdot)\) \(\chi_{485}(69,\cdot)\) \(\chi_{485}(139,\cdot)\) \(\chi_{485}(149,\cdot)\) \(\chi_{485}(164,\cdot)\) \(\chi_{485}(174,\cdot)\) \(\chi_{485}(214,\cdot)\) \(\chi_{485}(224,\cdot)\) \(\chi_{485}(239,\cdot)\) \(\chi_{485}(249,\cdot)\) \(\chi_{485}(319,\cdot)\) \(\chi_{485}(354,\cdot)\) \(\chi_{485}(369,\cdot)\) \(\chi_{485}(434,\cdot)\) \(\chi_{485}(439,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
Fixed field: | 32.0.5935315803327378381589507037815252283484449810801350950039360504150390625.1 |
Values on generators
\((292,296)\) → \((-1,e\left(\frac{17}{32}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 485 }(69, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) |
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)