![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(768, base_ring=CyclotomicField(64))
M = H._module
chi = DirichletCharacter(H, M([0,59,32]))
![Copy content]()
pari:[g,chi] = znchar(Mod(29,768))
| Modulus: | \(768\) | |
| Conductor: | \(768\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(64\) |
![Copy content]()
sage:chi.multiplicative_order()
![Copy content]()
pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
pari:#znconreyconductor(g,chi)==1
|
| Minimal: | yes |
| Parity: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
pari:zncharisodd(g,chi)
|
\(\chi_{768}(5,\cdot)\)
\(\chi_{768}(29,\cdot)\)
\(\chi_{768}(53,\cdot)\)
\(\chi_{768}(77,\cdot)\)
\(\chi_{768}(101,\cdot)\)
\(\chi_{768}(125,\cdot)\)
\(\chi_{768}(149,\cdot)\)
\(\chi_{768}(173,\cdot)\)
\(\chi_{768}(197,\cdot)\)
\(\chi_{768}(221,\cdot)\)
\(\chi_{768}(245,\cdot)\)
\(\chi_{768}(269,\cdot)\)
\(\chi_{768}(293,\cdot)\)
\(\chi_{768}(317,\cdot)\)
\(\chi_{768}(341,\cdot)\)
\(\chi_{768}(365,\cdot)\)
\(\chi_{768}(389,\cdot)\)
\(\chi_{768}(413,\cdot)\)
\(\chi_{768}(437,\cdot)\)
\(\chi_{768}(461,\cdot)\)
\(\chi_{768}(485,\cdot)\)
\(\chi_{768}(509,\cdot)\)
\(\chi_{768}(533,\cdot)\)
\(\chi_{768}(557,\cdot)\)
\(\chi_{768}(581,\cdot)\)
\(\chi_{768}(605,\cdot)\)
\(\chi_{768}(629,\cdot)\)
\(\chi_{768}(653,\cdot)\)
\(\chi_{768}(677,\cdot)\)
\(\chi_{768}(701,\cdot)\)
...
![Copy content]()
sage:chi.galois_orbit()
![Copy content]()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((511,517,257)\) → \((1,e\left(\frac{59}{64}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 768 }(29, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{3}{8}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
![Copy content]()
sage:chi.gauss_sum(a)
![Copy content]()
pari:znchargauss(g,chi,a)
![Copy content]()
sage:chi.jacobi_sum(n)
![Copy content]()
sage:chi.kloosterman_sum(a,b)
