from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(256, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,9]))
pari: [g,chi] = znchar(Mod(39,256))
Basic properties
| Modulus: | \(256\) | |
| 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 256.l
\(\chi_{256}(7,\cdot)\) \(\chi_{256}(23,\cdot)\) \(\chi_{256}(39,\cdot)\) \(\chi_{256}(55,\cdot)\) \(\chi_{256}(71,\cdot)\) \(\chi_{256}(87,\cdot)\) \(\chi_{256}(103,\cdot)\) \(\chi_{256}(119,\cdot)\) \(\chi_{256}(135,\cdot)\) \(\chi_{256}(151,\cdot)\) \(\chi_{256}(167,\cdot)\) \(\chi_{256}(183,\cdot)\) \(\chi_{256}(199,\cdot)\) \(\chi_{256}(215,\cdot)\) \(\chi_{256}(231,\cdot)\) \(\chi_{256}(247,\cdot)\)
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
\((255,5)\) → \((-1,e\left(\frac{9}{32}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 256 }(39, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) |
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)