from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(768, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,27,0]))
pari: [g,chi] = znchar(Mod(73,768))
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 \(\chi_{128}(29,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 768.v
\(\chi_{768}(25,\cdot)\) \(\chi_{768}(73,\cdot)\) \(\chi_{768}(121,\cdot)\) \(\chi_{768}(169,\cdot)\) \(\chi_{768}(217,\cdot)\) \(\chi_{768}(265,\cdot)\) \(\chi_{768}(313,\cdot)\) \(\chi_{768}(361,\cdot)\) \(\chi_{768}(409,\cdot)\) \(\chi_{768}(457,\cdot)\) \(\chi_{768}(505,\cdot)\) \(\chi_{768}(553,\cdot)\) \(\chi_{768}(601,\cdot)\) \(\chi_{768}(649,\cdot)\) \(\chi_{768}(697,\cdot)\) \(\chi_{768}(745,\cdot)\)
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
\((511,517,257)\) → \((1,e\left(\frac{27}{32}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 768 }(73, a) \) | \(1\) | \(1\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(-i\) |
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)