from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(512, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,5]))
pari: [g,chi] = znchar(Mod(49,512))
Basic properties
Modulus: | \(512\) | |
Conductor: | \(128\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{128}(53,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 512.k
\(\chi_{512}(17,\cdot)\) \(\chi_{512}(49,\cdot)\) \(\chi_{512}(81,\cdot)\) \(\chi_{512}(113,\cdot)\) \(\chi_{512}(145,\cdot)\) \(\chi_{512}(177,\cdot)\) \(\chi_{512}(209,\cdot)\) \(\chi_{512}(241,\cdot)\) \(\chi_{512}(273,\cdot)\) \(\chi_{512}(305,\cdot)\) \(\chi_{512}(337,\cdot)\) \(\chi_{512}(369,\cdot)\) \(\chi_{512}(401,\cdot)\) \(\chi_{512}(433,\cdot)\) \(\chi_{512}(465,\cdot)\) \(\chi_{512}(497,\cdot)\)
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: | \(\Q(\zeta_{128})^+\) |
Values on generators
\((511,5)\) → \((1,e\left(\frac{5}{32}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 512 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) |
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)