from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(512, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,9]))
pari: [g,chi] = znchar(Mod(15,512))
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 \(\chi_{128}(27,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 512.l
\(\chi_{512}(15,\cdot)\) \(\chi_{512}(47,\cdot)\) \(\chi_{512}(79,\cdot)\) \(\chi_{512}(111,\cdot)\) \(\chi_{512}(143,\cdot)\) \(\chi_{512}(175,\cdot)\) \(\chi_{512}(207,\cdot)\) \(\chi_{512}(239,\cdot)\) \(\chi_{512}(271,\cdot)\) \(\chi_{512}(303,\cdot)\) \(\chi_{512}(335,\cdot)\) \(\chi_{512}(367,\cdot)\) \(\chi_{512}(399,\cdot)\) \(\chi_{512}(431,\cdot)\) \(\chi_{512}(463,\cdot)\) \(\chi_{512}(495,\cdot)\)
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: | 32.0.3138550867693340381917894711603833208051177722232017256448.1 |
Values on generators
\((511,5)\) → \((-1,e\left(\frac{9}{32}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 512 }(15, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) |
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)