from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(768, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,31,0]))
pari: [g,chi] = znchar(Mod(727,768))
Basic properties
Modulus: | \(768\) | |
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 \(\chi_{128}(51,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 768.u
\(\chi_{768}(7,\cdot)\) \(\chi_{768}(55,\cdot)\) \(\chi_{768}(103,\cdot)\) \(\chi_{768}(151,\cdot)\) \(\chi_{768}(199,\cdot)\) \(\chi_{768}(247,\cdot)\) \(\chi_{768}(295,\cdot)\) \(\chi_{768}(343,\cdot)\) \(\chi_{768}(391,\cdot)\) \(\chi_{768}(439,\cdot)\) \(\chi_{768}(487,\cdot)\) \(\chi_{768}(535,\cdot)\) \(\chi_{768}(583,\cdot)\) \(\chi_{768}(631,\cdot)\) \(\chi_{768}(679,\cdot)\) \(\chi_{768}(727,\cdot)\)
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
\((511,517,257)\) → \((-1,e\left(\frac{31}{32}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 768 }(727, a) \) | \(-1\) | \(1\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(i\) |
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)