from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(680, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,8,4,7]))
pari: [g,chi] = znchar(Mod(317,680))
Basic properties
| Modulus: | \(680\) | |
| Conductor: | \(680\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | 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 680.ct
\(\chi_{680}(133,\cdot)\) \(\chi_{680}(173,\cdot)\) \(\chi_{680}(197,\cdot)\) \(\chi_{680}(317,\cdot)\) \(\chi_{680}(397,\cdot)\) \(\chi_{680}(413,\cdot)\) \(\chi_{680}(517,\cdot)\) \(\chi_{680}(573,\cdot)\)
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: | \(\Q(\zeta_{16})\) |
| Fixed field: | 16.16.11724484818984205488128000000000000.2 |
Values on generators
\((511,341,137,241)\) → \((1,-1,i,e\left(\frac{7}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 680 }(317, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(-i\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) |
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)