from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(776, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,16,5]))
pari: [g,chi] = znchar(Mod(531,776))
Basic properties
Modulus: | \(776\) | |
Conductor: | \(776\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 776.bm
\(\chi_{776}(19,\cdot)\) \(\chi_{776}(51,\cdot)\) \(\chi_{776}(67,\cdot)\) \(\chi_{776}(131,\cdot)\) \(\chi_{776}(139,\cdot)\) \(\chi_{776}(443,\cdot)\) \(\chi_{776}(451,\cdot)\) \(\chi_{776}(515,\cdot)\) \(\chi_{776}(531,\cdot)\) \(\chi_{776}(563,\cdot)\) \(\chi_{776}(627,\cdot)\) \(\chi_{776}(651,\cdot)\) \(\chi_{776}(659,\cdot)\) \(\chi_{776}(699,\cdot)\) \(\chi_{776}(707,\cdot)\) \(\chi_{776}(731,\cdot)\)
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: | 32.32.10948725162062396375893585077651059129669167570225987349215648531823007367168.1 |
Values on generators
\((583,389,393)\) → \((-1,-1,e\left(\frac{5}{32}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 776 }(531, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) |
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)