from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(896, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,23,16]))
pari: [g,chi] = znchar(Mod(531,896))
Basic properties
| Modulus: | \(896\) | |
| Conductor: | \(896\) | 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 896.bk
\(\chi_{896}(27,\cdot)\) \(\chi_{896}(83,\cdot)\) \(\chi_{896}(139,\cdot)\) \(\chi_{896}(195,\cdot)\) \(\chi_{896}(251,\cdot)\) \(\chi_{896}(307,\cdot)\) \(\chi_{896}(363,\cdot)\) \(\chi_{896}(419,\cdot)\) \(\chi_{896}(475,\cdot)\) \(\chi_{896}(531,\cdot)\) \(\chi_{896}(587,\cdot)\) \(\chi_{896}(643,\cdot)\) \(\chi_{896}(699,\cdot)\) \(\chi_{896}(755,\cdot)\) \(\chi_{896}(811,\cdot)\) \(\chi_{896}(867,\cdot)\)
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: | 32.32.104303243075213755167445035578915122359095224799654955003407693930037248.1 |
Values on generators
\((127,645,129)\) → \((-1,e\left(\frac{23}{32}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 896 }(531, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) |
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)