![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(736, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,18]))
![Copy content]()
pari:[g,chi] = znchar(Mod(57,736))
\(\chi_{736}(57,\cdot)\)
\(\chi_{736}(89,\cdot)\)
\(\chi_{736}(153,\cdot)\)
\(\chi_{736}(201,\cdot)\)
\(\chi_{736}(217,\cdot)\)
\(\chi_{736}(249,\cdot)\)
\(\chi_{736}(281,\cdot)\)
\(\chi_{736}(297,\cdot)\)
\(\chi_{736}(313,\cdot)\)
\(\chi_{736}(329,\cdot)\)
\(\chi_{736}(425,\cdot)\)
\(\chi_{736}(457,\cdot)\)
\(\chi_{736}(521,\cdot)\)
\(\chi_{736}(569,\cdot)\)
\(\chi_{736}(585,\cdot)\)
\(\chi_{736}(617,\cdot)\)
\(\chi_{736}(649,\cdot)\)
\(\chi_{736}(665,\cdot)\)
\(\chi_{736}(681,\cdot)\)
\(\chi_{736}(697,\cdot)\)
![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 ]
\((415,645,97)\) → \((1,i,e\left(\frac{9}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 736 }(57, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) |
![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)
