from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(128, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,1]))
pari: [g,chi] = znchar(Mod(5,128))
Basic properties
| Modulus: | \(128\) | |
| Conductor: | \(128\) | 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 128.k
\(\chi_{128}(5,\cdot)\) \(\chi_{128}(13,\cdot)\) \(\chi_{128}(21,\cdot)\) \(\chi_{128}(29,\cdot)\) \(\chi_{128}(37,\cdot)\) \(\chi_{128}(45,\cdot)\) \(\chi_{128}(53,\cdot)\) \(\chi_{128}(61,\cdot)\) \(\chi_{128}(69,\cdot)\) \(\chi_{128}(77,\cdot)\) \(\chi_{128}(85,\cdot)\) \(\chi_{128}(93,\cdot)\) \(\chi_{128}(101,\cdot)\) \(\chi_{128}(109,\cdot)\) \(\chi_{128}(117,\cdot)\) \(\chi_{128}(125,\cdot)\)
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
\((127,5)\) → \((1,e\left(\frac{1}{32}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 128 }(5, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) |
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)