![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(588, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,37]))
![Copy content]()
pari:[g,chi] = znchar(Mod(269,588))
\(\chi_{588}(5,\cdot)\)
\(\chi_{588}(17,\cdot)\)
\(\chi_{588}(89,\cdot)\)
\(\chi_{588}(101,\cdot)\)
\(\chi_{588}(173,\cdot)\)
\(\chi_{588}(185,\cdot)\)
\(\chi_{588}(257,\cdot)\)
\(\chi_{588}(269,\cdot)\)
\(\chi_{588}(341,\cdot)\)
\(\chi_{588}(353,\cdot)\)
\(\chi_{588}(425,\cdot)\)
\(\chi_{588}(437,\cdot)\)
![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 ]
\((295,197,493)\) → \((1,-1,e\left(\frac{37}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 588 }(269, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{21}\right)\) |
![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)
